Potential Theory on Minimal Hypersurfaces II: Hardy Structures and Schr\"odinger Operators

We extend the potential theory on almost minimzers from Part 1. We introduce so-called Hardy structures to study many classical operators using the tools from part 1. Furthermore, we show that for a naturally defined operator L, minimal growth of positive solutions of Lw = 0 towards the singular set is a stable property. It persists under perturbations or blow-ups of the underlying spaces. This is the key result to develop a dimensional induction scheme for the asymptotic analysis of these operators near the singular set.


Introduction
In this paper we further develop the potential theory on generally singular area minimizing and, more generally, on almost minimizing hypersurfaces we initiated in the first part of this work [L1]. We focus on elliptic operators naturally associate to such hypersurfaces, for instance, from the minimality constraint of the hypersurface and the geometry of the ambient space. This analysis can be applied to investigate the ambient space. A famous example hereof is the (non)existence of positive scalar curvature metrics on compact manifolds which is closely related to the potential theory of the conformal Laplacian on its minimal hypersurfaces [KW], [SY].
We know from [L2] that the S-uniformity of H \Σ, for an (almost) minimizing hypersurface H with singular set Σ suffices to establish a transparent potential theory on H \ Σ. This comes along with boundary Harnack inequalities H \ Σ relative the boundary Σ and a Martin theory saying that Σ is just the Martin boundary for a large class of operators, the so-called S-adapted operators.
In the present paper we take the applications of S-structures to the elliptic analysis on (almost) minimizers much further. We first show that (almost) minimizers actually carry an enhanced version of S-structures, namely a so-called Hardy S-structure. These satisfy an additional geometric coupling relation between S-structures and the curvature of almost minimizers. It follows that many classical elliptic operators, like the Jacobi field operator or the conformal Laplacian, are actually S-adapted so that our techniques from [L1] apply. Typically, these operators will be Schrödinger operators L = L(H) on H \ Σ, and the assignment H → L(H) commutes with blow-ups. That is, for infinite scalings m·H around any p ∈ Σ ⊂ H, L(m · H) converges to L(C) on any tangent cone C in p as m → ∞. For these so-called natural Schrödinger operators our theory works particularly nicely.
The question of understanding their ground states and Green's functions is of great importance. Towards the singularity set these characteristic solutions all have minimal growth compared with other solutions. This property is the starting point for our further analysis. Namely, we prove the allegedly plausible but rather non-trivial fact that minimal growth of solutions u > 0 on H \Σ of L(H) f = 0 towards some singular point p ∈ Σ persists under blow-ups.
In particular we find that, for any tangent cone C around p, solutions of L(C) f = 0 which are induced from u have minimal growth towards the singularity set Σ C of the cone. The Martin theory in [L1] shows that these solutions are unique up to constant multiples and they admit a separation of variables. From this observation we can start a scheme of inductive descent and continue with blow-ups in singular points of C \ {0} to get simpler cones of the form C * × R until we reach C • × R k , where C • is only singular at its tip. The resulting tree of blow-ups together with the induced solutions, analyzed using the individual potential theory on these blow-ups, allows a detailed analysis of solutions on H \ Σ near Σ.
As in [L2] this analysis does not use the structure of Σ itself. Instead it is naturally and tightly connected to S-structures and the properties of the associated hyperbolic unfoldings.

Basic Concepts and Notations
We recall some basic notations from the first part of this work, cf. [L1,Chapter 1.1] and [L2,Appendix A] for details.
In this paper H n denotes a connected integer multiplicity rectifiable current of dimension n ≥ 2 inside some complete, smooth Riemannian manifold (M n+1 , g M ). By Σ H (or simply Σ if there is no risk of confusion) we denote the set of singular points of H. It is known to have Hausdorff dimension ≤ dim H − 7. We also write σ C , in place of Σ C , for the singular set of a minimal cone C, in particular if we want to emphasize that C is viewed as a tangent cone.
The induced Riemannian metric on H will be denoted by g H . For λ > 0 we let λ · M or λ · H denote the conformally rescaled Riemannian manifolds (M, λ 2 · g) or (H, λ 2 · g H ). We refer to the induced distance function d = d H on H as intrinsic distance. Further, A = A H is the second fundamental form of H ⊂ M and |A H | its norm.
We shall consider the following basic subclasses of such currents. H c n : H n ⊂ M n+1 is compact locally mass minimizing without boundary.
H R n : H n ⊂ R n+1 is a complete hypersurface in flat Euclidean space (R n+1 , g eucl ) with 0 ∈ H, that is an oriented minimal boundary of some open set in R n+1 .
H n : H n := H c n ∪ H R n and H := n≥1 H n . We briefly refer to H ∈ H as an area minimizer.
C n : C n ⊂ H R n is the space of area minimizing n-cones in R n+1 with tip in 0.
SC n : SC n ⊂ C n is the subset of cones which are at least singular in 0.
K n−1 : For any area minimizing cone C ⊂ R n+1 with tip 0, we get the non-minimizing minimal hypersurface S C in the unit sphere and we set K n−1 := {S C | C ∈ C n }. We write K = n≥1 K n−1 for the space of all such hypersurfaces S C . G c n : H n ⊂ M n+1 is a compact embedded almost minimizer with ∂H = ∅. We set G c := n≥1 G c n .
G n : G n := G c n ∪ H R n and G := n≥1 G n . These are the main classes considered in this paper.
We denote the one-point compactification of a hypersurface H ∈ H R n by H. For the singular set Σ H of some H ∈ H R n we always add ∞ H to Σ as well, even when Σ is already compact, to define Σ H := Σ H ∪ ∞ H . On the other hand, for H ∈ G c n we set H = H and Σ = Σ.
We call an assignment A which associates with any H ∈ G a function A H : H \ Σ H → R an S-transform provided it satisfies the subsequent list of axioms.
(S1) Trivial Gauge If H ⊂ M is totally geodesic, then A H ≡ 0.
(S2) S-Properties If H is not totally geodesic, then the level sets A c := A −1 H (c), for c > 0, we call the |A|-skins, surround the level sets of |A|: Like |A H |, A H anticommutes with scalings, i.e., A λ·H ≡ λ −1 · A H for any λ > 0.
(S3) Lipschitz regularity If H is not totally geodesic, and thus A H > 0, we define the S-distance δ A H := 1/ A H . This function is L A -Lipschitz regular for some constant L A = L( A , n) > 0, i.e., |δ A H (p) − δ A H (q)| ≤ L A · d H (p, q) for any p, q ∈ H \ Σ and any H ∈ G n .
If H is totally geodesic, and thus A H ≡ 0, we set δ A H ≡ ∞ and |δ A H (p)−δ A H (q)| ≡ 0.
(S4) Naturality If H i ∈ H n , i ≥ 1, is a sequence converging* to the limit space H ∞ ∈ H n , then A H i C α −→ A H∞ for any α ∈ (0, 1). For general H ∈ G n , this holds for blow-ups: −→ A H∞ , for any sequence τ i → ∞ so that τ i · H → H ∞ ∈ H R n .
Remark 1. A construction of a concrete S-transform by merging the metric g H on H \ Σ H and the second fundamental form A = A H into one scalar function A H on H \ Σ H was given in [L2]. 2. The S-distance δ A is merely Lipschitz regular, but admits a Whitney type C ∞ -smoothing δ A * which satisfies (S1)-(S3) and which is quasi-natural in the sense that c 1 · δ A (x) ≤ δ A * (x) ≤ c 2 · δ A (x) for some constants c 1 , c 2 > 0, cf. [L2,Proposition B.3] for details.

Statement of Results
In [L1] we introduced S-adapted operators and studied their basic potential theory on almost minimizers H ∈ G. This class of operators is rich but a priori unrelated to the geometry of (H \ Σ, g H ). Further, it depends on the chosen S-transform A . In the first part of this paper we resolve this (apparent) issue. We show that there are S-transforms with a tighter coupling between |A| and A beyond the axioms (S1)-(S4). Then we use this to prove that many classical operators are S-adapted. Typically, these operators are symmetric and we are mostly interested in eigenvalue problems, so we shall focus on this situation. We recall the following definitions and results from [L1,Definition 1,Theorem 7 and 8]. For this we use special charts for H \ Σ, namely S-adapted charts. These are bi-Lipschitz charts ψ p : B R (p) → R centered in p ∈ H \ Σ, for some Lipschitz constant independent of p, and where the radius R of the ball is, up to some common constant, just 1/ A H (p), cf. [L1,Chapter 2.3] and [L2,Proposition B.1].
Definition Let H ∈ G. We call a symmetric second order elliptic operator L on H \Σ shifted S-adapted supposed the following two conditions hold: A -Adaptedness L satisfies S-weighted uniformity conditions with respect to the charts ψ p . Namely, we can write for some locally β-Hölder continuous coefficients a ij , β ∈ (0, 1], measurable functions b i and c, and there exists a k L = k ≥ 1 such that for any p ∈ H \ Σ and ξ ∈ R n : A -Finiteness There exists a finite constant τ = τ (L, A , H) > −∞ such that for any smooth function f which is compactly supported in H \ Σ, we have For any shifted S-adapted L we set Again, L λ is shifted S-adapted. Moreover, it is S-adapted if and only if λ < λ A L,H .
Coming back to the inequality A H ≥ |A H | we note that there cannot be a pointwise inverse inequality. For instance, singular cones may contain subcones where |A| ≡ 0 whereas A > 0. However, we can prove an inverse integral inequality for special S-transforms. To exclude the previous counterexample we need to include a gradient into the integrals. We obtain thus a generalization of both the Poincaré inequality and the sharper Hardy inequality for the Laplacian −∆ Eucl on flat Euclidean domains D ⊂ R n , to operators which couple to |A| on curved manifolds H \ Σ with boundary Σ, H ∈ G, cf.Ch.2.1.
Theorem 1 (Hardy S-Structures, see Theorem 2.1) There are S-transforms A such that for all H ∈ G and f ∈ C ∞ 0 (H \ Σ), the space of smooth functions compactly supported in H \ Σ, the following Hardy relations hold: (H c ) For any compact H ∈ G c and any smooth (2, 0)-tensor B on the ambient space M of H with B| H ≡ −A H there exists a constant k H;B > 0 such that For singular H ∈ G c , |A| is unbounded whereas |B| H | remains bounded, hence, in this case the condition B| H ≡ −A is redundant.
n we only consider the case B = 0. Then the Hardy constant depends solely on the dimension, that is, k H;0 = k n > 0, and we have These relations also apply to the case Σ = ∅ via the convention 1/dist(x, Σ) = 0. An Stransform satisfying both axioms (H) = (H c ) + (H R ) is called a Hardy S-transform.
Remark 1. The ambient field B incorporates geometric or physical constraints on M . An example is the second fundamental form of M in a still higher dimensional space. In this context, we consider hypersurfaces in H R n primarily as limit spaces under blow-ups, that is, infinite scaling of a given H ∈ G around its singularities. Since |B| λM = λ −1 · |B| M , for λ > 0, and λ → ∞ during the blow-up process, this suggests to focus on the case B = 0 on H ∈ H R n . Theorem 2 (Curvature Constraints, see Theorem 2.8) Let A be a Hardy S-transform. Further, let H n = H ∈ G with H ⊂ M n+1 and such that H \ Σ is non-compact and non-totally geodesic. The following operators are (shifted) S-adapted on H \ Σ:

(H c ) implies the Poincaré type inequality
Furthermore, if scal M ≥ 0 and H ∈ H, then L H is even S-adapted.
(ii) More generally, let S be any smooth function on M . Then the S-conformal Laplacian When H is compact, the principal eigenvalue λ A −∆,H vanishes and the ground state is given by a constant function. In particular, H \Σ has the Liouville property saying that all bounded harmonic functions are constant.
(iv) For any smooth (2, 0)-tensor B on the ambient space M with B| H ≡ −A if H is compact and B = 0 if H ∈ H R n , the A+B Laplacian C H;A,B := −∆+|A+B| H | 2 is an S-adapted operator.
These operators have two basic properties in common. First, they are naturally associated with any H ∈ G. This means that there is a unique expression for L(H) on H \ Σ H such that the assignment H → L(H) commutes with the convergence of sequences of almost minimizers, cf. Ch. 3.2 for details. Second, they are Schrödinger operators with finite principal eigenvalues. We merge these properties into one concept (cf. also Chapters 4.1 -4.2): Definition A natural and shifted S-adapted operator L is called a natural Schrödinger operator if for any given H ∈ H, L(H) has the form We take the analysis of natural Schrödinger operators near singular points beyond the Martin theory on H \ Σ H by considering tangent cones and cone reduction arguments. These are common in the geometric study of (almost) minimizers, for instance to prove bounds on the codimension of their singularity sets. For natural Schrödinger operators we build a matching analytic reduction scheme. In view of the key role played by the boundary Harnack inequalities [L1, Theorem 1 and 2] our goal is to understand how minimal growth towards singularities, transfers to the associated induced solutions on the tangent cones obtained by blowing up.
The minimal growth concept we use in this context is that of solutionsL-vanishing in (parts of) Σ H . We recall from [L1] that a solution u > 0 of L f = 0 in p ∈ Σ is L-vanishing in p when there exists a supersolution w > 0 with u/w(x) → 0 for x → p with x ∈ H \ Σ.
The following Theorem asserts an inheritance of minimal growth properties under blow-ups of the underlying spaces. The remarkable point is that we are comparing the fine asymptotic analysis of distinct spaces with completely different singularity sets. We are not aware of any comparable result in the literature. Remark If we do not fix the singular point while we scale H, that is, we consider a converging subsequence of pointed spaces (s i · H, p i ) for p i → p, p i ∈ Σ H , and s i → ∞ of scaling factors, then the limit space (H ∞ , p ∞ ) can be a general Euclidean hypersurface in H R n . In this case any induced solution on H ∞ \ Σ H∞ is L-vanishing along Σ H∞ .
The S-adaptedness of L(C) implies in particular that Martin theory applies to C \ σ C . By [L1,Theorem 3] there is exactly one Martin boundary point Ψ + at ∞. We emphasize that it is not the symmetry of the cone but the S-uniformity of C which implies uniqueness of Ψ + . Indeed, Ancona gave striking counterexamples of Euclidean cones over non-uniform spherical domains with uncountable families of minimal Martin boundary points at infinity [An3]. In turn, the uniqueness of Ψ + combined with the cone symmetry of C and the separation of variables for natural Schrödinger operators over cones yields the following structure result.
Theorem 4 (Separation of Variables, see Theorem 4.4) Let C ∈ SC n and L be a natural Schrödinger operator. For the S-adapted operator L λ = L − λ · A 2 · Id, λ < λ A L,C , and the two distinguished points Ψ − at zero and Ψ + at infinity in the Martin boundary of L λ (C), we have, in terms of polar coordinates (ω, r): These results describe the behavior of the Ψ ± on individual cones. In general, however, we have infinitely many distinct tangent cones around a singular point p ∈ Σ H . Our next result is a variant of Theorem 3 asserting that the assignment of Ψ ± to the underlying cones is natural in the following sense.
Theorem 5 (Naturality of Ψ ± , see Theorem 3.9) Let L be a natural and S-adapted Schrödinger operator on cones C ∈ SC n . Then for any flat norm converging sequence C i → C ∞ , i → ∞, with suitably normalized associated solutions Ψ ± (C i ) and Ψ ± (C ∞ ), we have where ID is the asymptotic identification representing C i as a smooth section of the normal bundle over C ∞ , cf. Chapter 3.1 and [L2,Chapter 1.3].
A more general version of this result applies to converging sequences in H R n , cf. 3.10.
Remark Theorems 3, 4 and 5 give us a recipe for the asymptotic analysis of natural Sadapted Schrödinger operators near Σ ⊂ H. For instance, if p ∈ Σ is an isolated point and its tangent cones are singular only in the tip, these results entail a sharp description of the solutions of minimal growth on H \ Σ near p. On the other hand, we can treat more complicated singular sets with tangent cones singular also outside the tip as follows: We consider the tangent cone and an induced solution of minimal growth towards σ C ⊂ C as the new initial object. Now we blow-up in points of σ C distinct from the tip and iterate this process until we reach the elementary case of a product cone R m × C n−m , for some C n−m ⊂ R n−m singular only in 0 (this happens at the latest after dim H − 7 times). Since by uniqueness, the induced minimal solutions are R n−m -translation symmetric we end up with the explicit description provided by Theorem 4 over cones singular only at the tip. Finally, Theorem 5 takes care of the non-uniqueness of tangent cones and yields uniform control for all these cones. We can then work backwards this tree of blow-ups to the initially given H ∈ H from such a terminal node R m × C n−m , for some C n−m ⊂ R n−m singular only in 0.
As an example (and with later applications in scalar curvature geometry and general relativity in mind) we derive more detailed results for the conformal Laplacians L H and L C on H and its tangent cones C: Theorem 6 (Conformal Laplacians, see Theorem 4.5) There are constants Λ n > λ n > 0 depending only on n such that for λ ∈ (0, λ n ] and any singular area minimizing cone C, (L C ) λ is S-adapted. Furthermore, we have the following estimates for Ψ ± (ω, r) = ψ(ω) · r α ± : • |ψ| L 1 (S C \Σ S C ) ≤ a n,λ · inf ω∈S C \Σ S C ψ(ω) for some constant a n,λ > 0 depending only on n and λ.
An interesting point in this result is the uniform separation of the exponents given by the lower bound α + − α − ≥ 3/4 · (n − 2).

Hardy S-Structures
The goal of this chapter is to show that many classical operators are actually shifted S-adapted with respect to a special subclass of S-structures which we call Hardy S-structures. Similar to Martin theory for S-adapted operators, this is rather a property of the underlying space than of the operators.

Hardy Inequalities
The Poincaré inequality is a frequently used tool in geometric analysis. For a Lipschitz regular and bounded Euclidean domain D ⊂ R n it asserts that there is some constant a D > 0 such that for any smooth function φ compactly supported in D. The Hardy inequality is a remarkable refinement of this result. Indeed, under the previous assumptions we can even find a constant c D > 0 such that for any such φ, cf. the detailed expositions [BEL] and [GN] for some background information.
Now for H ∈ G, the singular set Σ ⊂ H plays the role of a boundary of H \ Σ, and we shall prove similar Hardy inequalities as in the Euclidean case using the metric distance dist(x, Σ). We refer to these as metric Hardy inequalities. More importantly, we get for suitable S-transforms a stronger S-Hardy inequality using the S-distance δ A (x). In fact, the metric Hardy inqualities are a simple byproduct of our S-formalism, in the same way as uniformity of H \ Σ follows from S-uniformity [L2].
We get two versions of Hardy inequalities according to whether H is compact or lies in H R n . The latter case is the main case as it covers blow-up limits of hypersurfaces in G around singular points.
Theorem 2.1 (Hardy S-Structures) There are S-transforms A such that for all H ∈ G and f ∈ C ∞ 0 (H \ Σ), the space of smooth functions compactly supported in H \ Σ, the following holds.
(H c ) For any compact H ∈ G c and any smooth (2, 0)-tensor B on the ambient space For singular H ∈ G c , |A| is unbounded but |B| H | remains bounded, so that, in this case, the condition B| H ≡ −A is redundant.
(H R ) For H ∈ H R n we only consider the case B = 0. Then the Hardy constant depends solely on the dimension, that is, k H;0 = k n > 0, and we have These relations also apply to the case Σ = ∅ via the convention 1/dist(x, Σ) = 0. An Stransform satisfying both axioms (H) = (H c ) + (H R ) is called a Hardy S-transform.
Remark 2.2 For compact H ∈ G, we need both integrands |∇f | 2 and |A + B| H | 2 on the left hand side of these inequalities. Indeed, since H is compact and codim Σ ≥ 2, the coarea formula (cf. for instance [GMS,Theorem 2 On the other hand, |A| usually vanishes or at least converges to zero along suitable sequences of points approaching Σ, while A converges to +∞. (This corresponds to rays in the tangent cones along which the cones are totally geodesic). Then, for smooth functions supported around such points, we get Since the Hardy inequalities (H c ) and (H R ) do not follow from the S-axioms we have to revisit our construction of metric S-transforms A α,H which we briefly recall. If H ∈ G is totally geodesic, we set A α,H ≡ 0. Otherwise, for c > 0 we set A c (α) := the boundary of the A key input in proving that A α also satisfies (H) = (H c ) + (H R ) will be [L1,Lemma A.7]. This is, roughly speaking, a quantitative version of the fact that tangent cones in singular points are also singular.
Remark 2.4 Apart of the proof of Proposition 2.3 our later applications shall only make use of the axioms (S1)-(S4) and (H), but not of any particular feature of A α . Once we have proven this result we therefore simply add (H) to our set of S-axioms and henceforth assume that all our S-transforms are Hardy S-transforms.
For the proof of Proposition 2.3 we may assume that H is not totally geodesic, whence A α > 0. Otherweise H is totally geodesic, so A α (x) ≡ 0, and the Hardy inequalities hold trivially. We also note that the largest constant k = k α,H such that holds for any f ∈ C ∞ 0 (H \ Σ) is nothing but the first eigenvalue λ Pα of the weighted operator Since A α is locally Lipschitz, elliptic theory shows that eigenfunctions of P α are C 2,γ -regular for any γ ∈ (0, 1), cf. [GT,Chapter 6.4]. Then λ Pα can be written as a Rayleigh quotient To estimate λ Pα we localize the problem to Neumann eigenvalues on regular balls. Then we take covers by such balls with controlled covering numbers and use them to derive a positive lower estimate for the eigenvalue λ Pα .

Neumann Eigenvalues on Balls
The Neumann eigenvalues of P α on balls are scaling invariant: For any ball B r (p) ⊂ H \ Σ, r > 0 and scaling factor µ > 0 we have for the integrands of the Rayleigh quotient (2) |∇f | 2 + |A + B| H | 2 (numerator) and A 2 (denominator) rescale by the same factor µ −2 > 0.
Next we would like to establish a lower bound for ν α (B r (p)). However, there is no uniform positive lower bound for r → 0 if |A|(p) = 0. Conversely, when the balls become too large, we can neither control their geometry nor understand the eigenvalues or the covering numbers. On the other hand, when we approach Σ we get better and better local approximations by Euclidean hypersurfaces in H R n (after rescaling to a unit size). They are singular and real analytic hypersurfaces in R n . This leads us to the idea to let A α determine the radius of the balls when we are close to Σ. Towards this end we notice that A α (x) ≥ α/dist(x, Σ) means that B α/ A α (p) (p) ∩ Σ = ∅ for any p ∈ H \ Σ. For these balls we have the following estimates: Lemma 2.5 For any α ∈ (0, 1] and µ ∈ (0, 1/2) there is a neighborhood U α,µ of Σ as well as a constant ζ(U α,µ ) > 0 such that Proof To simplify notation we only consider the case α = 1. We will make explicit use of the definition of A 1 . Let us assume that there is a sequence of points in view of the definition of A , the boundedness of |B| and |B| λM = λ −1 · |B| M , λ > 0. Moreover, taking without loss of generality p ∞ = 0, we get a subsequence of the pointed spaces ( A 1 (p i ) · H, p i ) which converge compactly in flat norm and thus in C k -norm, k ≥ 0, to (H ∞ , 0) ⊂ (R n+1 , 0) by standard regularity theory. (We generally use k = 5 to control also second derivatives of curvatures.) We notice that and thus A 1 (0) = 1. Here where we use the following inequality [L2,Lemma A.7]: For any µ ∈ (0, 1], there is a constant c(µ, n) > 0 such that We claim that the first Neumann eigenvalue Note that the tensor B vanishes in the limit since as Since there is a positive upper bound for A 1 over B µ (0) ∩ H ∞ , it suffices to consider the usual non-weighted Neumann eigenvalue we would have a smooth positive function u with ∆u = |A| 2 · u and vanishing normal derivative along ∂B µ (0) ∩ H ∞ . But then Stokes theorem would imply Bµ(0)∩H∞ ∆u = 0, whereas Bµ(0)∩H∞ |A| 2 · u > 0. Hence for sufficiently large i,

Controlled Covers
To derive Proposition 2.3 we combine these estimates for Neumann eigenvalues with the following S-adapted covers for H ∈ H introduced in [L2, Proposition B.1] to construct the S-Whitney smoothings (which equally apply to H ∈ G since the whole argument is based on blow-up arguments to limits in H R ). For a consistent statement we also include the trivial case of totally geodesic H ∈ G.
Proposition 2.6 (S-Adapted Covers) For any H ∈ G and size parameter ξ ∈ (0, ξ 0 ) for some ξ 0 (n, L A α ) ∈ (0, 1/(10 3 ·L A α )) we get a locally finite cover A = {B Θ(p) | p ∈ A} of H \Σ by closed balls of radius Θ(p) := ξ/ A α (p) = ξ · δ A α (p) and some discrete set A ⊂ H \ Σ, such that for a suitably small neighborhood Q of Σ we have: In particular, for any z ∈ Q and ρ ∈ (0, 10) the covering number by balls centered in A ∩ Q is uniformly bounded. Such a cover A will be called S-adapted. These covers have the following properties.
Proof of Proposition 2.3 We first consider (H c ). We start with the simplest case where H is totally geodesic. Then A α ≡ 0 and the Hardy inequality becomes trivial. Thus we take H not totally geodesic so that A α > 0 on H.
Assuming that H is regular we have an upper bound for A α > 0 on H. It is therefore enough prove the positivity of the usual eigenvalue of −∆ + |A + B| H | 2 . Now |A + B| H | 2 ≥ 0, and in some open set it is positive since B| H ≡ −A. Thus, for any smooth positive function u (including the first eigenfunction) we have H ∆u = 0, whereas H |A + B| H | 2 · u > 0. Hence, the eigenvalue of −∆ + |A + B| H | 2 cannot be zero. Now we turn to the main case where H is singular. As above we notice that ν α (H \ W ) > 0 for sufficiently small and smoothly bounded neighborhoods W of Σ ⊂ H. Indeed, H \ W contains a non-empty open ball where |A + B| H | > 0 if W is sufficiently small, since there are p k ∈ H \ Σ where |A|(p k ) ≥ k, for any k ≥ 1, whereas |B| H | remains bounded. For fixed W we have positive bounds for A α , and we can consider the standard Neumann eigenfunction. Again we can invoke Stokes' Theorem to infer that ν α (H \ W ) > 0. We note in passing that once we found a neighborhood W with ν α (H \ W ) > 0, positivity continues to hold for all neighborhoods W * of Σ with W * ⊂ W . However, this argument does not give a uniform positive lower bound while W * shrinks to Σ since A α diverges when we approach Σ.
Next we take an S-adapted cover for ξ = µ · α and radii Θ(p) = µ · α/ A α (p), and we choose W ⊂ Q, so that A Q = {B Θ(p) | p ∈ Q} is covering of W with covering number c(n). To ease notation we will write B(p) = B µ·α/ A α (p) (p). Then for any f ∈ C ∞ 0 (H \ Σ), f ≡ 0 we get the following estimate: On the other hand, the Neumann eigenvalues ν α (B(p)) are uniformly bounded from below by Put differently the Hardy inequality holds for k α,H := λ Pα .
These arguments apply equally well in the case H ∈ H R . In this case Proposition 2.6 asserts that we get S-adapted covers not only of small neighborhoods of Σ, but of the entire hypersurface. Hence the previous chain of inequalities now yields Therefore, the Hardy inequality holds for any H ∈ H R and for k α,n := ζ/c(n).

Geometric Operators
For the remainder of this paper we fix a Hardy S-transform A on G. In this section we shall employ the Hardy axiom (H) to verify the S-adaptedness of some basic geometric operators.
The natural geometric operator which we call the A+B Laplacian, couples to the second fundamental form of H ⊂ M and an additional (2, 0)-tensor B on M . Of course, this definition is suggested right from the definition of Hardy S-transforms, and the operator will be useful to provide lower bounds for variational integrals involving other S-adapted operators.
Proof Theorem 2.1 shows the A -finiteness of C H;A,B for a positive τ > 0. Thus we only need to verify the A -adaptedness. With respect to the charts ψ p we write We recall that in local coordinates the Laplacian ∆ u equals 1 and that |A + B| H | 2 is smooth. Hence the coefficients a ij , b i , c are smooth. Since the charts ψ p are the geodesic coordinates around p we have with respect to these charts a ij (p) = δ ij , b i (p) = 0 and c(p) = |A+B| H | 2 (p). Moreover, |B| H | remains bounded. Thus, the unfolding correspondence [L1,Proposition 3.3] shows that δ 2 A * ·L is an adapted operator on (H \Σ, d A * ).
One source for S-adapted operators are geometric and physical variational problems. Here, geometric properties of the ambient space like its curvature, or a given tensor T coming from physical constraints, can translate into S-adapted operators on the hypersurface.
(ii) More generally, let S be any smooth function on M . Then the S-conformal Laplacian If H is compact, the principal eigenvalue λ A −∆,H vanishes and the ground state is given by a constant function. In particular, H \Σ has the Liouville property saying that all bounded harmonic functions are constant.
(iv) The Jacobi field operator Proof By the arguments of Lemma 2.7, the operators are adapted to A .
To show S-adaptedness it remains to consider the validity of the Hardy inequality. Towards this end consider the Gauß-Codazzi equation where trA H is the mean curvature of H. If H is minimal, then trA H = 0. The terms 2Ric M (ν, ν) and scal M remain bounded, whereas |A H | 2 and scal H diverge when we approach Σ on H. Also S| H remains bounded. Therefore, We now turn to the S-conformal Laplacian L H,S . As an area minimizing hypersurface H is also stable, that is, the second variation of the area functional is non-negative. Hence, if f is a smooth function on H with supp f ⊂ H \ Σ and ν the outward normal vector field over H \ Σ, then The Gauß-Codazzi equation (6) gives an equivalent formulation of For the Laplacian the condition λ A −∆,H > −∞ is obvious since the variational integral is just the Dirichlet integral H |∇f | 2 dV ≥ 0. If H is compact, it is easy to see that the principal eigenvalue equals 0. Here we apply again the coarea formula, cf. [GMS,Theorem 2.1.5.3], and use the fact that the codimension of Σ is ≥ 2. Every constant function v solves ∆ v = 0. Thus v ≡ 1 can be taken as the ground state for compact H. Moreover, v := u + inf H\Σ u + 1 is a positive harmonic function for any bounded harmonic function u. Hence v, and therefore u, are constant functions.
Finally, we consider the Jacobi field operator J H . From the minimality of H and (7), for any smooth function f on H with supp f ⊂ H \ Σ.

Blow-Up Martin Theory
The structure of the singularity set of almost minimizers does not directly transfer to their tangent cones. However, for S-adapted operators which are naturally associated with minimizers we do have a non-trivial relationship between their Martin theories on the minimizer and their tangent cones. In particular, by our main result the minimal growth properties of solutions towards singularities are inherited by all tangent cones. This plausible result uses virtually any structural detail of our theory, from S-uniformity of almost minimizers to boundary Harnack inequalities of S-adapted operators. are complete manifolds and the H n i are almost minimizers with basepoint p i ∈ H i . Furthermore, we assume that as i → ∞, the pointed manifolds (M i , p i ) are compactly C k -converging, for some k ≥ 5, to a pointed limit manifold (M, p). Hence, for any given R > 0 and sufficiently large i ≥ i R we have diffeomorphisms

ID-Maps and Tangent Cones
We assume that ∂H i ∩ B R (p i ) = ∅.
• Basic compactness results show that the H i subconverge in flat norm to a limit area minimizer H ⊂ M containing p, that is, as For instance, consider an initial hypersurface H 0 in M 0 . A sequence τ i → ∞ gives rise to the rescaled sequence H i := τ i · H 0 ⊂ M i := τ i · M 0 of blow-ups. Fix an H 0 -singular point p 0 ∈ Σ ⊂ H 0 and set p i := p 0 . Then M i converges compactly to R n+1 while H i subconverges in flat norm to a (n area minimizing) tangent cone H ⊂ R n+1 .
• When B R (p) ∩ H is smooth, standard regularity shows the following. First, flat norm convergence of Ψ i (B R (p i ) ∩ H i ) to B r (p) ∩ H implies that B R (p i ) ∩ H i is also smooth for sufficiently large i. Further, we obtain C k -convergence in the following sense: The and Γ i C k -converges to the zero section which we identify with B R (p) ∩ H.
Definition 3.1 (ID-map) For sufficiently large i we call the uniquely determined section the asymptotic identification or ID-map for short.
Put differently, the regularity theory of area minimizers implies that we can approximate smooth regions of H by corresponding (and also smooth) regions of H i through the canonical local diffeomorphisms given by ID-maps, cf. also [L2,Chapter 1.3]. Moreover, ID-maps are arbitrarily close to isometries in C k -topology for sufficiently large i. ID-maps are useful to visualize situations where we compare functions or operators on different underlying hypersurfaces in G, for instance when passing to blow-up limits.
This naturality concept readily extends to more general assignments H → F H of tensors or operators defined over H \ Σ. In the latter case we refer to the operator F (H) itself as natural. It becomes an important problem to determine which properties of natural operators are stable under convergence, for instance under blowing up. As a first example we prove that (shifted) S-adaptedness is stable. Proof For any smooth function f with compact support K ⊂ N \ Σ N we choose a sufficiently large scaling factor γ ≫ 1 such that the ID-map between γ · H and N in a neighborhood of K is very close to an isometry in C 5 -topology. Then the naturality of L and A implies that upon choosing γ large enough. Thus the eigenvalue of δ 2 A · L(H) on H is a lower estimate for the second integral. In particular, the Hardy inequality holds for L(N ) on N . The adaptedness property of L(N ) follows from the scaling invariance of the estimates and the adaptedness of L(H).
Tangent Cone Freezing [L1,Ch.4.1] Let H ∈ G. Blowing up around p ∈ Σ H yields only converging subsequences. In particular, there is no canonical tangent cone which could serve as a local model. Nevertheless, we can approximate the local geometry near z using tangent cones. For this, we take ω > 0 and consider the S-pencil pointing to p ∈ Σ defined by In view of scaling arguments it is also useful to consider the truncated S-pencil While we zoom into some given singular point p ∈ H, formally accomplished by scaling with increasingly large τ > 0, we observe that τ · TP H (p, ω, R/τ, r/τ ) is better and better C kapproximated by the corresponding truncated S-pencil in some (usually changing) tangent cone. Intuitively, the twisting of P(p, ω) slows down as τ → ∞ until it asymptotically freezes and ressembles a cone-like geometry.
This can be concisely expressed in terms of ID-maps ID = ID τ,R,r,ω : TP C τ p (0, ω, R, r) → τ · TP H (p, ω, R/τ, r/τ ) satisfying |ID − id C τ p | C 5 (TP(0,ω,R,r)) ≤ ε for the norm on sections of the normal bundle, thinking of id C τ p as the zero section of the normal bundle over C τ p .

Induced Solutions
The S-freezing of Proposition 3.4 shows that the ID-maps are almost isometric maps between suitable domains with compact closure in H \ Σ H and arbitrarily large truncated pencils TP C τ p (0, ω, R, r) ⊂ C τ p , where R > 0 is arbitrarily large and r, ω > 0 is arbitrarily small.
Next we consider a natural operator L. Restricted to these truncated pencils, the coefficients of L(C τ p ) are arbitrarily close in Hölder norm to those of the ID-pull-back of L(H).
Thus, for a positive solution u of L(H) f = 0, u • ID solves an elliptic equation whose coefficients are arbitrarily close in Hölder norm to those of L(H). Since weak solutions of L f = 0 are C 2,α -regular, we get locally uniform constants for elliptic regularity estimates and Harnack inequalities for positive solutions, cf. [BJS], [E] or [GT].
Now we take a sequence τ i → ∞ as i → ∞. The embedding C 2,β ⊂ C 2,α is compact on bounded domains for β ∈ (0, α). Hence, after normalizing the value of the locally defined almost-solutions u • ID of L(C τ i p ) f = 0 in some basepoint of TP C τ i p (0, ω, R, r) ⊂ C τ i p , there is a subsequence compactly C 2,β -converging to a positive solution on TP C τ∞ p (0, ω, R, r) ⊂ C τ∞ p as i → ∞. This also uses Harnack inequalities away from the basepoint.
We observe that, while i → ∞, the ID-map becomes more and more isometric on evergrowing truncated pencils TP C τ i p (0, ω, R, r) ⊂ C τ i p as R → ∞ and r → 0, eventually exhausting C τ i p \ σ C τ i p in the limit. Therefore, possibly after selecting further subsequences, this process induces a positive C 2,β -regular function v solving L(C τ i p \ σ C τ i p ) f = 0 on the whole of C τ i p \ σ C τ i p , that is, we obtain an entire solution. This gives rise to the following functional version of Proposition 3.4.
Proof The first assertion is merely Proposition 3.4. Next, assume that for some ε > 0, R > r > 0, ω > 0, there exists a sequence τ i → ∞ such that for any entire solution v > 0 of Compactness of the space of tangent cones T p in p gives a subsequence τ i k such that C τ i k p converges to some tangent cone C p . Hence for any entire solution v > 0 of L(C p ) f = 0 on C p \ σ Cp we would also have that However, our discussion before the proposition shows that we can choose another subsequence τ i km of τ i k such that the normalized u•ID induces a positive entire solution on C p contradicting (10).
Corollary 3.6 (Asymptotic Functional Freezing on Cone Spaces) Let α ∈ (0, 1), ε, ω > 0 and R ≫ 1 ≫ r > 0 with sufficiently large R and small r. Then there exists a ζ(L, ε, ω, R, r) > 0 such that following holds. For any C and C ′ ∈ C n with d H (S C , S C ′ ) ≤ ζ we have |ID − id C | C 5 (TP C (0,ω,R,r)) ≤ ε, and any entire solution Completely similarly we get such entire solutions on more general blow-up hypersurfaces in H R n obtained from scaling an almost minimizer H ∈ G without fixing a basepoint.
Definition 3.7 (Induced Solutions) We call such a positive entire solution on the limit hypersurfaces in H R n an induced solution.
Since induced solutions are positive entire solutions we can employ the full Martin theory on spaces in H R n and their S-adapted operators to analyze them.

Minimal Growth Stability
Let L be a natural S-adapted Schrödinger operator L. We show that L-vanishing is a stable property which persists under deformations of the underlying spaces. A far reaching consequence is that any positive solution of L f = 0 which is L-vanishing around some singular p ∈ Σ H induces solutions on any tangent cone C at p which are L-vanishing towards the entire singular set σ C . In other words, the asymptotic minimal growth of solutions and the convergence of the underlying spaces are interchangeable. This is a remarkable phenomenon as the singular set of the underlying space, and thus the meaning of L-vanishing, may change dramatically under deformations.
Theorem 3.8 (Blow-Up Stability of L-Vanishing Properties) Let H ∈ G and L be some natural, S-adapted Schrödinger operator on H \ Σ. Further, let u > 0 be a solution of L f = 0 which is L-vanishing in a neighborhood V of some point p ∈ Σ. Then we have the following inheritance results: • If C is a tangent cone of H in p, then any solution induced on C is L-vanishing along σ C .
• More generally, for a sequence s i → ∞ of scaling factors and a sequence of points p i → p in Σ H such that (s i · H, p i ) subconverges to a limit space (H ∞ , p ∞ ) with H ∞ ∈ H R n , any induced solution on H ∞ \ Σ H∞ L-vanishes along Σ ∞ . The Martin theory for these limit spaces says that there is, up to multiples, precisely one positive solution L-vanishing along the singular set Σ H∞ . It is the unique and minimal Martin boundary point at infinity ∞ H∞ cf. [L1,Theorem 3]. We write this solution as Ψ + = Ψ + (H ∞ , L). Similarly, for any cone C ∈ SC n , there is also a unique (minimal) Martin boundary point at the origin 0 C , the positive solution Ψ − = Ψ − (C, L) which is L-vanishing along σ C ∪ {∞ C } \ {0 C }.
A basic problem with tangent cones in singular points is that they are generally nonunique. The following second stability theorem partial compensates this issue. We show that the induced solutions change continuously when we move from one to another tangent cone. Therefore we can extend compactness results for area minimizers to the assigned functions Ψ + . This helps to derive a variety of uniform estimates for such solutions for all cones in SC n .
Theorem 3.9 (Stable L-Vanishing on Cones) Let L be a natural and S-adapted Schrödinger operator on cones C ∈ SC n . Then, for any flat norm converging sequence C i → C ∞ , i → ∞, with appropriately normalized associated solutions Ψ ± (C i ) and Ψ ± (C ∞ ), we have A more general version of this result applies to H R n . A particularly interesting special case are degenerating sequences of regular Euclidean hypersurfaces H i ∈ H R n with singular limit. Then the induced solutions are always L-vanishing along the singular set of the limit hypersurface.
Theorem 3.10 (Stable L-Vanishing on H R n ) Let H i ∈ H R n be a compactly converging sequence, i ≥ 1, with limit H ∞ ∈ H R n . Further, let L = −∆ + V be a natural, S-adapted Schrödinger operator with λ The proofs of 3.8 -3.10 occupy the remainder of this section. The arguments for these results essentially coincide and will be addressed simultaneously. We first derive variants 3.12 of these results for minimal Green's functions which are L-vanishing along the entire singular set. This way we separate the invariance of L-vanishing properties from the problem to work with the localization that the given solution u > 0 of L f = 0 is L-vanishing only in a neighborhood U of some point p ∈ Σ. This localization is considered in a second step. This is another occasion where we employ the boundary Harnack inequalities already used to ensure the uniqueness of Ψ ± .
In these arguments we use some functional analytic consequences of S-adaptedness. We show that the Hardy inequality extends from C ∞ 0 (H \ Σ) to H 1,2 A (H \ Σ) and that furthermore, S-adaptedness also allows us to control the H 1,2 A -norm, see [L1,Ch.5.1].
Proof We recall that A -adaptedness means that for some a L > 0 the potential V satisfies Therefore, [L1,Theorem 5.3] shows that the inequality H\Σ f ·Lf dV ≥ λ A L,H · H\Σ A 2 ·f 2 dV not only holds for test functions in C ∞ 0 (H \ Σ) but actually for all functions in H 1,2 A (H \ Σ). This implies the first inequality (12) on H 1,2 A (H \ Σ).
For the second inequality we use (14) again to write whence (13)  The subsequent theorem asserts that minimality of Green's functions G(x, y) is stable under perturbation of the underlying area minimizer. In the argument we show that G(x, y) can be described as finite energy minimizers of a Dirichlet type integral outside regular balls B ρ (x) centered around the pole x of G(x, ·). By the previous lemma this also entails finite H 1,2 A -norm. This is by no means obvious since in general, even minimal growth solutions strongly diverge towards Σ and general solutions usually lead to an infinite H 1,2 A -norm, cf. Theorem 4.4 and Proposition 4.5 below for growth estimates.
We consider two situations of converging hypersurfaces with possibly non identical ambient spaces as described in Ch.3.1 and using ID-map identifications: (S 1 ) Let H i ∈ H R n , i ≥ 1, be compactly converging, with basepoints a i ∈ H i \ Σ H i and possibly with Σ H i = ∅. Let H ∞ ∈ H be the limit with basepoint a ∞ = lim i→∞ a i , d(a ∞ , 0) = dist(a ∞ , Σ H∞ ) = 5 and 0 ∈ Σ H∞ ⊂ H R n . Further, let L = −∆ + V be a natural, S-adapted Schrödinger operator with λ A L,H i ≥ c for some c > 0 independent of i.
(S 2 ) Let H ∈ G c and H i = τ i · H ∈ G c , for some sequence τ i → ∞. Also we choose basepoints a i ∈ H \ Σ H , i ≥ 1. We assume the H i and a i converge to some limit H ∞ ∈ H R n with basepoint a ∞ = lim i→∞ a i , d(a ∞ , 0) = dist(a ∞ , Σ H∞ ) = 5 and 0 ∈ Σ H∞ ⊂ H R n . Further, let L = −∆ + V be a natural, S-adapted Schrödinger operator on H.
The reason why we only consider the case of blow-ups of one given almost minimizer H ∈ G c is to ensure uniform control over the convergence and naturality properties of A .
Proposition 3.12 (Stability of Minimal Green's Functions) We assume we are in one of the two situations S 1 and S 2 . We denote by G i = G i (·, a i ) > 0 the minimal Green's function on H i with pole in a i and normalized to ∂B 1 (a i ) G i = 1, where i ≥ 1 or i = ∞.
Then there is a subsequence, which we still denote G i , such that as i → ∞, Proof We focus on case S 1 . Case S 2 then follows along similar lines noting that the scaling invariance of the A 2 -weighted principal eigenvalue implies the condition λ A L,H i ≥ c for some constant c > 0 from the S-adaptedness of L.
It is a trivial fact that the limit of the G i is again a Green's function; the point is to show its minimality. The idea is to characterize a minimal Green's function G(·, p) on H, outside some neighborhood U of the pole p ∈ H \ Σ H , as the minimizer of the following Dirichlet type integral S-adaptedness of L implies then that the minimality of the J-integral is preserved under convergence of the underlying hypersurfaces.
In our situation where we have a compactly converging sequence of pointed minimizers H i → H ∞ with a i → a ∞ we set U i = ID(B 1 (a ∞ )) and U ∞ = B 1 (a ∞ ). Without loss of generality we may also assume that B 5 (a i ) ⊂ H i \ Σ H i , since B 5 (a ∞ ) ⊂ H i \ Σ H∞ , and these balls are, possibly after rescaling, diffeomorphic and uniformly almost isometric in C 3 -norm to the Euclidean ball B 5 (0). The proof will be divided into three steps. In the first two preliminary steps we derive results needed in the third and main step. In the first step we show that the functionals J H i \U i , i ≥ 0 or i = ∞, are uniformly bounded. In Step 2 we shall prove that the G i are the unique minimizers of the Dirichlet type integral above. In Step 3 we prove that G i converge to a limit solution G * which minimizes the Dirichlet integral on H ∞ \ U ∞ . (The uniqueness from Step 2 then implies that G * = G ∞ .) For this we prove that there is no bubbling-off of negative contributions to J H i \U i (f ) before we reach the limit. This will use the S-adaptedness of L and the fact that H 1,2 A (H \ Σ) ≡ H 1,2 A ,0 (H \ Σ) proved in [L1,Theorem 5.3].
Step 1 For the infima inf For the lower bound let f be a smooth function with supp f ⊂ H i \ Σ H i and f | ∂U i = G i . We extend f as a smooth function to F i on U i . Hence |J U i (F i )| ≤ c for some uniform constant c < ∞. The Hardy inequality for L entails, by Lemma 3.3, that On the other hand, a fixed test function f supported in B 2 (a ∞ ) gives a uniform upper bound for J H∞\U∞ (f ) on H ∞ and thus, via almost isometric ID-map pull-back for sufficiently large i, also for J H i \U i (f ) on H i .
Step 2 For the Dirichlet problem of J H i \U i (·) over H 1,2 A (H i \ Σ H i ) the minimal Green's function G i restricted to H i \U i is the unique minimizer v > 0 with boundary value φ = G ∞ •ID on ∂U i . Remark 3.13 1. If we work with functions in H 1,2 A , then the Dirichlet problem is considered in the trace sense, cf. [E,Chapter 5.5 and 6.1]. At any rate, the resulting minimizers are regular and thus solve the problem in the classical sense.
2. The claim follows from the general theory of symmetric semi-bounded operators cf. Remark 3.14 below. But the following explicit argument may be more satisfactory.
For the uniqueness part, we assume we had two distinct positive solutions Regularity theory shows that v 1,2 are at least C 2,αregular. Upon relabeling v 1 and v 2 we can find an η < 1 such that η · v 1 and v 2 coincide in a non-empty set C outside B 1 (p). The Hopf maximum principle shows that C must be a smooth submanifold where the graphs of v 1 and v 2 intersect transversally. Since both functions minimize J we observe that Thus min{η · v 1 , v 2 } or max{η · v 1 , v 2 } also minimizes the functional J and must be smooth. But this contradicts the fact that both min{η · v 1 , v 2 } and max{η · v 1 , v 2 } are non-smooth along C. This proves the uniqueness assertion.
For the existence part we choose a sequence of smoothly and compactly bounded domains There is a unique solution Q i,m for the Dirichlet problem with boundary data Q i,m ≡ G i on ∂U and Q i,m ≡ 0 on ∂D m which minimizes the Dirichlet functional. We infer from [L1,Theorem 5.3] and the proof of the critical operator case in [L1,Theorem 5.7 Thus the unique minimizer of J H i \U i is the minimal Green's function.
For any η > 0, comparing the functional over compactly supported test functions on the limit space H ∞ with the functional over compactly supported test functions on H i via ID-maps yields an i η with This, in turn, implies that cf. (13) of Lemma 3.11. c * > 0 is a common upper bound for the contribution of the extensions of the G i to U i by bounded regular functions. By regularity theory the G i compactly C 2,αsubconverge to some solution G * > 0 with G * ∈ H 1,2 A (H ∞ \(U ∞ ∪ Σ H∞ )) and G * = G ∞ on ∂U .
In view of the compactly C 2,α -subconverging G i , this amounts to exclude bubbling-off phenomena of negative contributions to the Dirichlet integral concentrating near Σ i which are no longer visible in the limit. Towards this end, we let I ε := {x ∈ H i \ Σ H i | δ A (x) < ε}, for ε > 0 which we choose sufficiently small so that U i ∩ I 3·ε = ∅ and R ≥ 10. Assume that there is a constant c > 0 independent of both, ε and R, such that there is an i 0 ε.R so that To argue that this cannot happen we use the setup from the proof of [L1,Theorem 5.3]. We first construct a cut-off function concentrated near Σ i by taking some fixed ψ ∈ C ∞ (R, [0, 1]) with ψ ≡ 1 on R ≤0 and ψ ≡ 0 on R ≥1 . Then we set Since δ A is Lipschitz continuous there exists some constant c(ψ) > 0 depending only on ψ such that for the distributional derivative From (17) and (18) we see that Secondly, we consider a cut-off towards infinity by setting ψ R (x) := ψ(R −1 · |x| − 1) for x ∈ H ∞ \ Σ ∞ and R ≥ 10. We note as a counterpart of (18) that . This means that for some b H∞ > 0. Then we have using (19) and (20): We observe, from G * ∈ H 1,2 A (H ∞ \(U ∞ ∪Σ H∞ )), that |G * | H 1,2 A (I 2·ζ \I ζ ) and |G * | H 1,2 tend to 0, when ε → 0 respectively R → ∞. Transferring the cut-off region via the ID-map to H i shows that for sufficiently small ε > 0, large R and i, the contributions to the two norms on the cut-off region become arbitrarily small: But this contradicts the S-adaptedness of L for test functions in H 1,2 Remark 3.14 The considered domain of a shifted S-adapted operator L is typically a space of (sufficiently) regular functions. For such domains the operators are symmetric on the original space H \ Σ, but in general not self-adjoint. (Note in passing that the associated operator δ −2 A · L on the hyperbolic unfolding is usually no longer a symmetric operator.) However, right from the definition, any shifted S-adapted operator L is a symmetric semibounded operator on C ∞ 0 (H \ Σ). This is sufficient to ensure that L admits a canonical selfadjoint extension L F , the so-called Friedrichs extension. Its domain D(L F ) satisfies For these Friedrichs extensions we have a general existence and uniqueness theory for the Dirichlet problem in Step 2 of Proposition 3.12 above, cf. [Hf,Ch.4.3] and [Z,Ch.5.5].
The boundary Harnack inequality allows us to derive the following localized versions of this stability of minimal Green's functions.
Proposition 3.15 (Localized Inheritance of L-Vanishing) We assume situation S 1 or S 2 . Then, let u i > 0 be a solution of L f = 0 on each H i which via ID-maps are compactly converging to some entire solution u ∞ > 0 of L f = 0 on H ∞ . If there is a ball B in a common ambient manifold of the H i and H ∞ around a singular point p ∈ Σ H∞ such that u i , for any i, is L-vanishing along B ∩ Σ H i , then u ∞ is also L-vanishing along B ∩ Σ H∞ .
The proof is done in three steps. In Step 1 we prepare canonical Φ δ -chains on the H i converging to a canonical Φ δ -chains on H ∞ we use as a neighborhood basis of 0. Next, in Step 2, we observe that for these Φ δ -chains on the H i we get common constants in the BHP on all H i , cf. [L1,Ch.2.1]. In Step 3 we use the stability of minimal Green's functions G i on the H i 3.12. Using Step 2, we see that the minimal Green's function G i upper bounds the u i on B ∩ H i by a common multiple of G i and this carries over to the limit. Thus, we infer that u ∞ is also L-vanishing along B ∩ Σ H∞ .

Remark 3.16 For
Step 1 we recall from [L1,Ch.3.1] that the BHP for S-adapted operators is a direct consequence of Ancona's BHP from the unfolding correspondence. For converging sequences H i as in situations S 1 and S 2 the associated Gromov hyperbolic (H i , d A H i ) converge compactly to that of the limit space H ∞ . For the Whitney smoothed S-metrics d A * , used in the unfolding correspondence, any such sequence (H i , d A * H i ) has subsequences converging to a Whitney smoothing (H ∞ , d A * H∞ ). This readily follows from the proof of [L1,Proposition B.3] since the basepoints of the S-adpated covers compactly subconverge. In what follows we therefore assume that the ( Proof of 3.15 Step 1 We first recall the construction of canonical Φ δ -chains from [L1,Lemma 2.4]. For a δ-hyperbolic space we consider geodesic arcs γ : (0, c) → X, for c > 10 3 · δ.
In both situations S 1 and S 2 we now turn to the Whitney smoothed hyperbolic unfoldings. We can choose hyperbolic geodesic arcs γ[ H∞ ) representing the Gromov boundary point 0 ∈ Σ H∞ . The convergence is formalized via IDmaps. This also implies that the canonical Φ δ -chains N δ k (γ[a i ]) ⊂ H i , of length m(i) → ∞, we assign to each of the γ[a i ], compactly converge to N δ k (γ[a ∞ ]) ⊂ H ∞ , via ID-maps.
Step 2 We can now apply the BHPs [L1,Theorem 3.4 and 3.5]. For a minimal Green's function G and solutions v > 0 of L f = 0 on H i \ Σ H i which is L-vanishing along N δ 2 (γ[a i ]) we have: For k L ≤ κ and ε L ≥ η the constant C only depends on κ and η > 0. In our cases this means that we can choose a common Harnack constant C for all i. For S 1 this from the stable BHP on H R n [L1,Theorem 3.5]. It gives the same constant for all such hypersurfaces. For S 2 we can employ the BHP on G n of [L1,Theorem 3.4] since we are working only with one fixed hypersurface.
Step 3 In both situations, S 1 or S 2 , we have u i ≤ C · G i on N δ 3 (γ[a i ]) for some common C ≥ 1. Now we recall from [BHK,Proposition 8.10 forms a neighborhood basis of 0 ∈ H ∞ . Since Σ H∞ ∩ B is compact, we can therefore assume (without loss of generality) that B ∩ H ∞ ⊂ N δ k (γ[a ∞ ]), for k = 1, ...3, i ≥ 1 and i = ∞.
Thus we also have This result also concludes the proof of 3.8 and 3.9 taking increasingly larger balls B in 3.15.

Martin Theory on Minimal Cones
Here we refine our analysis of natural Schrödinger operators. The minimal growth stability and Martin theory can be used to build an inductive asymptotic analysis of solutions with minimal growth near singular points using iterative blow-ups. In the cases of the Jacobi field operator and the conformal Laplacian we derive some further details.

Schrödinger Operators and Scaling Actions
For Schrödinger operators L which are naturally associated with H ∈ G we get a neat representation for certain distinguished solutions of L f = 0. We estimate their radial growth in the cone case to gain a detailed understanding of the growth of the Martin kernel for compact H.  For an area minimizing cone C ⊂ H the naturality implies that V C (t · x) = t −2 · V C (x) for any x ∈ C \ σ C and t > 0. In this case we can write where L × = L × (S C ) is an operator on S C . We will see in 4.4 below that L × is also a natural Schrödinger operator.
Scaling Actions On an area minimizing cones equipped with a natural and S-adapted operator L, both, the operator and solutions of L f = 0 reproduce themselves under scalings of the cone up to constant multiples. Concretely, we express L on C = R ≥0 × S C in geodesic coordinates x 1 = r, x 2 , ..., x n such that x 2 , ..., x n locally parametrize S C = C ∩ ∂B 1 (0): When L is natural this means that for any η > 0, Thus, for any function u(x) that solves L f = 0 its rescaling u(η · x) also solves this equation by the chain rule. In particular, the Green's function and the set of minimal solutions of L f = 0 are reproduced up to multiples under composition with the scaling map More concretely, we consider the map u → u • S η and normalize the values of the resulting functions u • S η to 1 in some base point p ∈ C \ σ C . In this way we define a scaling action S * η on the Martin boundary.

Lemma 4.3 (Attractors and Fixed
Points of S * η ) Let C ∈ SC n be a singular area minimizing cone and L be a natural S-adapted operator on C. Then we have: (i) The scaling action S * η on ∂ M (C, L) has exactly two fixed points: the tip 0 C and the point at infinity ∞ C , both viewed as minimal functions.
Proof This readily follows from the way the pole of G(·, S η (z)) shifts under these scaling operations.
Now we give a description of the two fixed point solutions in ∂ M (C, L). Despite the fact mentioned above that there is no proper way of transforming the Martin theory from the original hypersurface H to its tangent cones this will allow us to build an inductive decomposition scheme for solutions on H.
Theorem 4.4 (Separation of Variables) For any cone C ∈ SC n and any natural Schrödinger operator L let us consider the S-adapted operator L λ = L − λ · A 2 · Id, λ < λ A L,C . Then we have: • Viewed as functions Ψ − = 0 C and Ψ + = ∞ C on C \ σ the two fixed points 0 C , ∞ C ∈ ∂ M (C, L λ ) can be written as defined on S C \ Σ S C is a natural Schrödinger operator with non-weighted principal eigenvalue µ C,L × λ > −( n−2 2 ) 2 and ground state ψ(ω) > 0, that is, Proof We proceed in three steps: First, we show that Ψ ± can be written as a product Ψ ± (ω, r) = ψ ± (ω) · r α ± . Then we determine ψ ± and α ± for some inner approximation C by regular subcones. Finally, we prove that the resulting values converge to ψ ± and α ± on C.
Product Shape We first restrict the two fixed points Ψ ± ∈ ∂ M (C, L λ ) to a regular ray which we view as a restriction of a function f : R >0 → R >0 . Now up to a constant Ψ ± reproduces under scalings: For any η > 0, there is constant c η > 0 such that From this it follows that f is a monomial, that is, f (x) = a · x b for some constants a(v) > 0, b(v) ∈ R. This argument applies to any regular ray Γ v .
Next we consider the Harnack inequality for L λ on a ball B 2R (v) ⊂ C \ σ C for some R > 0. We get, for any positive solution u of L λ f = 0, the Harnack inequality for some constant independent of u. The crux of the matter is that both the scaling symmetry of C and the naturality of L λ imply that the same constant c can still be used in the Harnack inequality after scalings around the tip 0: For any s 0 we have u.
We reinsert Ψ ± (ω, r) = ψ ± (ω) · r α ± into the equation L λ f = 0 which we write in polar coordinates as in (22). A separation of variables shows that the ψ ± solve the equations Further, L × λ is again a natural Schrödinger operator and adapted to the S- Inner Regular Approximation We use an approximation by Dirichlet eigenvalue problems to show that • ψ − = ψ + and ψ := ψ ± is the corresponding ground state of L × λ . Towards this end we choose smoothly bounded domains We consider the positive solutions of L λ f = 0 on the cone C(D i ) ⊂ C over D i with vanishing boundary value along ∂C(D i ) \ {0}. Now we apply [L1,Remark 3.10]  Again, we insert Ψ ± [i](ω, r) into the equation L λ f = 0, written in polar coordinates as in (22) and find that the ψ ± [i] solve Over D i we apply the spectral theory for bounded domains and observe that the positive eigenfunctions ψ ± [i] must equal the uniquely determined first Dirichlet eigenfunction ψ[i] for the first eigenvalue µ[i] of L × λ . Thus we have whence µ[i] ≥ −(n − 2) 2 /4. Further, the variational characterization of these eigenvalues gives , since the space of admissible test functions on D i is a subset of the corresponding function space over D i+1 .
After a suitable normalization the first Dirichlet eigenfunctions ψ[i] of L × λ on D i converge C 3 -compactly on S C \ Σ S C as i → ∞ to a positive eigenfunction ψ * with eigenvalue µ * = lim i→∞ µ[i] ≥ −(n − 2) 2 /4. We also have the limits α * ± := lim i→∞ α ± [i]. From this we observe as in [L1,Theorem 5.7] that ψ * > 0 is the non-weighted ground state of L × λ on S C \ Σ S C for the eigenvalue µ * > −∞. Moreover, L × λ , and hence L × , are shifted S-adapted. Namely, A × > c S C > 0 and thus the principal eigenvalue of δ 2 A × · L × λ remains finite as the non-weighted principal eigenvalue µ * of L × λ is finite. Therefore, L × λ is a natural Schrödinger operator.

Geometric Operators on Cones
Next we focus on a subclass of natural Schrödinger operators which one typically encounters in applications to scalar curvature geometry. For these we will derive quite explicit growth estimates.
Theorem 4.5 (Eigenvalue Estimates for J C and L C ) Let C ∈ SC n be a singular area minimizing cone. Then we get the following estimates for the Jacobi field operator J C and the conformal Laplacian L C : • The principal eigenvalue λ J C ,C of J C is non-negative and • There are constants Λ n > λ n > 0 depending only on n such that (L C ) λ is S-adapted for any λ ≤ Λ n . Furthermore, for any λ ≤ λ n and thus α + ≥ −(1 − 3/4) · n − 2 2 , α − ≤ −(1 + 3/4) · n − 2 2 for λ ≤ λ n .
Next let λ ∈ (0, λ n ]. Since scal C ≤ 0 and the codimension of Σ S C is greater than two, we can find a function f ∈ C ∞ 0 (S C \ Σ S C ) equal to 1 outside a sufficiently small neighborhood of Σ S C , and equal to 0 close to Σ S C so that Due to • the naturality of S-transforms, and • the compactness of the space of all singular area minimizing cones C ⊂ R n+1 in flat norm topology and in compact C 5 -topology outside the singular sets, we know that there is a common positive lower bound on S C A | 2 S C dV for all such C. More precisely, there are constants η λ , ϑ λ > 0 which depend only on λ > 0 and n such that µ C,(L C ) × λ < −η λ < 0, whence 0 > −ϑ λ > α + and α − > ϑ λ − (n − 2) > −(n − 2) for any singular area minimizing cone C.
Inductive cone reduction arguments are a powerful tool to reduce general questions to low dimensions. In essence, this is an iterative blow-up process where we first blow-up around some p 0 ∈ Σ H and get a tangent cone C 1 . While scaling around 0 ∈ σ C 1 merely reproduces C 1 , blowing up around a singular point p 1 = 0 ∈ σ C 1 generates an area minimizing cone C 2 which can be written as a Riemannian product C 2 = R × C 3 for a lower dimensional area minimizing cone C 3 . This way we encounter cones C n ⊂ R n+1 which can be written as a Riemannian product R n−k × C k , where C k ⊂ R k+1 is a lower dimensional area minimizing cone singular at 0. A famous instance of this technique is Federer's estimate for the codimension of the singularity set.
For applications to scalar curvature geometry which we shall consider elsewhere, it is useful to establish this kind of cone reduction arguments for the conformal Laplacian. In this case we observe that the minimal function Ψ + (ω, r) of (L C n ) λ on C n shares the R n−k -translation symmetry with the underlying space R n−k × C k since Ψ + (ω, r) is uniquely determined. Hence Ψ + (ω, r)| {0}×C k satisfies the equation where L C k ,n := −∆ − n − 2 4(n − 1) · |A| 2 for n ≥ k and (L C k ,n ) λ = L C k ,n − λ · A 2 .
Thus L C k ,n is a dimensionally shifted version of the conformal Laplacian on the cone C k : The entities ∆, |A| 2 and A 2 are intrinsically defined on C k , while the dimensional shift comes from using n−2 4(n−1) in place of k−2 4(k−1) . The next two results describe the analysis of these operators.
Proposition 4.6 (Dimensionally Shifted L C ) Let C k ⊂ R k+1 be a singular area minimizing cone and n ≥ k. Then we have: • L C,n is S-adapted and for its principal eigenvalue there exists a uniform lower bound for all k-dimensional cones by a positive constant Λ * k > 0 independent of n.
Complementary to the previous discussions where we mostly focussed on the radial growth rate, we now describe some global properties of the spherical component ψ C (ω) of Ψ ± [n, k](ω, r) = ψ C (ω) · r α ± which is defined over S C \ Σ S C . In particular, this will yield uniform estimates for the radial growth near the singular set.
Although S C ⊂ ∂B 1 (0) is not a global area minimizer, it is an almost minimizer and it shares the regularity theory with proper area minimizers. We can also locally apply the Bombieri-Giusti Harnack inequality [BG,Theorem 6 p. 39] in the following form: For any superharmonic function w > 0 defined on the regular region of B R (x) ∩ S C for a sufficiently small extrinsically measured radius R > 0, we have 0 < 1 V ol n−1 (S C ∩ B r (x)) w p 1/p

≦ C · inf
Br(x) w for r ≦ β n · R and p < k−1 k−3 with constants C = C(S C , p) and β n > 0. We apply this to a finite cover B r (p j ), j = 1, ...m, of Σ S C by sufficiently small balls with B R (p j ) ⊂ I(ρ).
Since the complement in S C \ Σ S C of these open balls is compact, inf S C \Σ S C ψ C (ω) > 0 and |ψ C | L p (S C \Σ S C ) < ∞. Moreover, we obtain for each individual cone C the (trivial) estimate sup ω∈E(ρ) ψ C (ω) ≤ b · |ψ C | L 1 (S C \Σ S C ) for some suitably large b = b C,n,k,λ,ρ > 0. Since the ψ C are unique up to a multiple, the compactness of SC k , the naturality of |A| and A , and the standard elliptic theory for (L C,n ) × λ imply for all C k ∈ SC k the existence of some common a n,k,λ > 0 such that |ψ C | L 1 (S C \Σ S C ) ≤ a n,k,λ · inf ω∈S C \Σ S C ψ C (ω).
The assertion that |v| L q (B 1 (0)∩C k ) < ∞ for q < k k−2 and a solution v > 0 of (L C k ,n ) λ f = 0 follows completely similarly by invoking again the Bombieri-Giusti Harnack inequality.