Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch's comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li-Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch's comparison theorem, the proofs mainly rely on the first variation formula, thus are valid for general varifolds.


Introduction
Many inequalities that relate the mean curvature of submanifolds with other geometric quantities such as the diameter can be obtained in some way from monotonicity identities, which are formulas that can be used to deduce monotonicity of weighted density ratios. In the Euclidean case, these identities are typically proven by testing the first variation formula with certain vector fields. One of the main ingredients in the construction of these vector fields is the inclusion map of the submanifold into the ambient Euclidean space. A key observation in the computations is that its relative divergence equals the dimension of the submanifold. In the Riemannian case, the inclusion map of a submanifold is not a vector field; however one can perform analogous arguments by using the vector field r∇r, where r is the distance function to a given point (see for instance Anderson [3]). Indeed, its relative divergence is not constant but can be bounded below on small geodesic balls by Rauch's comparison theorem (see Lemma 2.7). Such an idea revealed to be very fruitful; for instance, it enabled Hoffman-Spruck [11] to derive a Sobolev inequality for Riemannian manifolds. The idea of testing the first variation formula with the vector field r∇r in combination with Hessian comparison theorems for the distance function that give a lower bound of the relative divergence was used again in the works of Karcher-Wood [14] and Xin [44]. Their resulting monotonicity inequalities imply Liouville type vanishing theorems for harmonic vector bundle valued p-forms. Later, the same idea was used by several authors to prove vanishing theorems in various settings, see for instance Dong-Wei [8]. The technique was recently applied by Mondino-Spadaro [26] to derive an inequality that relates the radius of balls with the volume and area of the boundary. See also Nardulli-Osorio Acevedo [27] who used the technique to prove monotonicity inequalities for varifolds on Riemannian manifolds. A weighted monotonicity inequality was obtained by Nguyen [28].
In the present paper, we apply the described technique to prove Riemannian counterparts of the Euclidean monotonicity inequalities and their consequences from Simon [38,39] and Allard [2]. Our main result is a general Li-Yau inequality (see Theorem 1.7). We start with a brief introduction to intrinsic varifolds on Riemannian manifolds in Section 1.1. All our monotonicity inequalities (see Section 3) as well as our main theorems (see Section 1.3) are valid for general varifolds. In particular, all our results can be applied to isometrically immersed Riemannian submanifolds (see Example 2.4). Indeed, the Li-Yau inequality for ambient manifolds with positive upper bound on the sectional curvature is also new in the smooth case.

Varifolds on Riemannian manifolds
Let m, n be positive integers satisfying m ≤ n. Given any n-dimensional vector space V , we define the Grassmann manifold G(V, m) to be the set of all m-dimensional linear subspaces of V . For V = R n , we write G(n, m) := G(R n , m). One can show that G(n, m) is a smooth Euclidean submanifold, see for instance [9, 3.2.29(4)].
Let (N, g) be an n-dimensional Riemannian manifold. We denote with G m (T N ) the Grassmann m-plane bundle of the tangent bundle T N of N . That is, there exists a map π : G m (T N ) → N such that for each p ∈ N , the fibre π −1 (p) is given by the Grassmannian manifold G(T p N, m). Given any open set U in N and a chart x : U → R n of N , we note that π −1 [U ] is homeomorphically mapped onto an open subset of R n × G(n, m) via With an m-dimensional varifold in N we mean a Radon measure V over G m (N ). The space of all m-dimensional varifolds on N is denoted with V m (N ). The weight measure V of a varifold V is defined by It is the push forward measure of the varifold under the projection G m (N ) → N . In particular, V is a Radon measure on N (see [22,Lemma 2.6]). The space of compactly supported vector fields on N is denoted with X (N ). Given any X ∈ X (N ), p ∈ N , and T ∈ G(T p N, m) with orthonormal basis {e 1 , . . . , e m }, we let where ∇ denotes the Levi-Civita connection. Moreover, we denote with spt X the support of X. The first variation of a varifold V is defined as the linear functional δV : X (N ) → R, δV (X) = div T X(p) dV (p, T ).
The total variation δV of δV is defined by δV (U ) = sup{δV (X) : X ∈ X (N ), spt X ⊂ U, g(X, X) ≤ 1} whenever U is an open subset of N , and Finally, we say that H is the generalised mean curvature of V in (N, g), if and only if H : N → T N is V measurable, δV is a Radon measure over N , there exists a δV measurable map η taking values in T N such that δV almost everywhere, g(η, η) ≤ 1, and The m-dimensional lower density Θ m * (µ, p) and upper density Θ * m (µ, p) of µ at p ∈ X are defined by The support spt µ of the measure µ is defined by Similarly, we say that the open ball B r (p) with radius r > 0 is a geodesic ball if it is a geodesically star-shaped neighbourhood of p.
Given a complete Riemannian manifold N , p ∈ N , and writing r = d(p, ·), we denote with Cut(p) the cut locus of p in N , and define the radial curvature K r on N \ Cut(p) to be the restriction of the sectional curvature to all planes that contain the gradient ∇r of r. Moreover, given any subset A ⊂ N , we denote with i p (A) the injectivity radius at p in N restricted to the subset A.
Typically, we denote with | · | g the norm induced by a Riemannian metric g.

Geometric inequalities
The following lower diameter bound for closed minimal submanifolds was proven and discussed by Xia [43] for the special case where the ambient manifold N is given by the n-dimensional real projective space of curvature 1. To the author's knowledge, little is known in the general case. In particular, the study of sharp lower bounds for different model spaces remains an open problem. The theorem is a direct consequence of Lemma 4.1 in combination with (2.9) and Example 2.4. (Lower diameter bounds for closed minimal Riemannian submanifolds).

Theorem
If instead, the sectional curvatures of N are pinched between 1 4 b and b, and N is simply connected, then, by a result of Klingenberg [16], (1.2) becomes In the special case where N = RP n is the n-dimensional real projective space of curvature 1, the lower bound in (1.4) is attained if and only if M is totally geodesic, see [43]. However, the same does not hold for the unit sphere N = S 7 . Indeed, the complex projective space CP 2 ( 4 3 ) of complex dimension 2 and complex sectional curvature 4 3 has diameter √ 3π 2 and can be isometrically and minimally imbedded into S 7 , see [43,34]. It remains an interesting open problem to study sharp lower diameter bounds for closed minimal submanifolds in different ambient manifolds, in particular in S n .
We have the following generalisation of (1.3) to asymptotically non-positively curved ambient manifolds as a consequence of Lemma 4.3 in combination with (2.9), and Example 2.4.

Theorem
Then, the extrinsic diameter d ext (M ) of M in N can be bounded below by the cut locus distance: In [39, Lemma 1.1], Simon showed a diameter pinching theorem for closed surfaces in the Euclidean space in terms of their Willmore energy and their area. The upper diameter bound was improved and generalised to Euclidean submanifolds by Topping [40] using the Michael-Simon Sobolev inequality [24]. It was further generalised to Euclidean submanifolds with boundary by Menne-Scharrer [22] leading to a priori diameter bounds for solutions of Plateau's problem. The Riemannian equivalent of Topping's upper diameter bound was proven by Wu-Zheng [42] using the Hoffman-Spruck Sobolev inequality [11]. The following theorem is the Riemannian version for varifolds of Simon's diameter pinching. It is proven in Section 4.1. In the smooth case it follows from [42].

Theorem (Diameter pinching).
Suppose n is an integer, n ≥ 2, N is a complete n-dimensional Riemannian manifold with positive injectivity radius i > 0, b ≥ 0, the sectional curvature satisfies K ≤ b, V ∈ V 2 (N ) has generalised mean curvature H, H is square integrable with respect to V , δV is absolutely continuous with respect to V , spt V is compact and connected,

5)
and and the extrinsic diameter d ext (spt V ) of the support of V is bounded above: In the early 60s, Willmore [41] showed that the energy now bearing his name is bounded below by 4π on the class of closed surfaces Σ ⊂ R 3 : where H denotes the sum of the principal curvatures and µ is the canonical Radon measure on Σ given by the immersion. The inequality is also referred to as Willmore inequality. Equality holds only for round spheres. Willmore's inequality was improved by Li-Yau [19, Theorem 6] for smoothly immersed closed surfaces f : Σ → R n : If there exists p ∈ R n with f −1 (p) = {x 1 , . . . , x k } where the x i 's are all distinct points in Σ, in other words f has a point of multiplicity k, then In particular, if the Willmore energy 1 4 Σ |H| 2 dµ lies strictly below 8π, then f is an embedding. Because of this property, the Li-Yau inequality has become very useful for the minimisation of the Willmore functional and, more generally, for the study of immersed surfaces. Due to the conformal invariance of the Willmore functional observed by Chen [7], Willmore's inequality has an analogue for surfaces Σ in the three sphere S 3 : and an analogue for surfaces Σ in the hyperbolic space H 3 : where |Σ| = 1 dµ denotes the area of Σ. Kleiner [15] showed for minimisers Σ 0 of the isoperimetric profile in a complete one-connected 3-dimensional Riemannian manifold without boundary and with sectional curvatures bounded above by b ≤ 0. Then, Ritoré [30] showed that ( Notice also the recent generalisation of the Willmore inequality to higher dimensional submanifolds by Agostiniani-Fogagnolo-Mazzieri [1]. They showed that for closed codimension 1 submanifolds M in a non-compact n-dimensional Riemannian manifold (N, g) with non-negative Ricci curvature, there holds where AVG(g) denotes the asymptotic volume ratio of (N, g). See also Chen [6] for the earlier Euclidean version.
In our following theorem, we upgrade the result of Schulze [37] about the inequality (1.13) to a Li-Yau inequality in any non-positively curved ambient manifold. More importantly, we upgrade the spherical Willmore inequality (1.11) to a Li-Yau inequality for varifolds on any Riemannian manifold with an upper bound (possibly positive) on the sectional curvature. The proof is done in Section 3.1. For non-positively curved ambient manifolds, it is analogous to Chai [5]; for ambient manifolds with positive upper bound on the curvature it is inspired by Simon [39].

Theorem (Li-Yau inequality). Suppose n is an integer
Then, there holds

Remark.
Notice that the existence of the density Θ 2 ( V , p) is part of the statement. Indeed, existence of the density as well as its upper semi-continuity are local statements that do not require any global upper bounds on the curvature nor do they require positive injectivity radius, see Theorem 3.6.
If the varifold is given by a smoothly immersed surface, then the theorem reads as follows (see Example 2.4). Then, the following two statements hold.

If b ≤ 0 and the image of f is contained in a geodesically star-shaped open neighbourhood of the multiplicity point, then
In particular, if the left hand side is strictly smaller than 8π, then f is an embedding.
1.10 Remark. The upper bound on the radius of the geodesic ball containing the image of f in (1) can be enlarged up to π √ b at the cost of a constant larger than 16 π 2 in front of the area. Moreover, in view of Lemma 3.1 and (3.5), the constant 16 π 2 cannot be expected to be sharp. It is an interesting open question whether or not the constant 16 π 2 can be replaced by 1. In view of the spherical version (1.11) this seems possible.
If N is a Cartan-Hadamard manifold, then N itself is a geodesically star-shaped open neighbourhood of any point. In particular, there is no condition on f in (2).
Isoperimetric inequalities play an important role in the theory of varifolds and its applications, see for instance [2,20,22]. They can be derived from Sobolev inequalities. Allard [2, Theorem 7.1] proved a Sobolev inequality for general varifolds in Euclidean space with a constant depending on the dimension of the varifold and the dimension of the ambient space. Michael-Simon [24] proved a Sobolev inequality for generalised submanifolds in Euclidean space where the constant depends only on the dimension of the submanifold. Its proof was later adapted by Simon [38, Theorem 18.6] for varifolds whose first variation is absolutely continuous with respect to the weight measure. These varifolds correspond to submanifolds without boundary. Menne-Scharrer [23] proved a general Sobolev inequality accounting for the unrectifiable part of the varifold as well. Finally, Hoyos [12,13] generalised Simon's version [38,Theorem 18.6] for varifolds on Riemannian manifolds. Our following theorem generalises his inequality for arbitrary varifolds whose first variation doesn't have to be absolutely continuous with respect to the weight measure. In this way, our resulting isoperimetric inequality indeed recovers the smooth version of Hoffman-Spruck [11, Theorem 2.2]. The proof of the following theorem can be found in Section 5.1.

Theorem (Sobolev inequality). Suppose m, n are positive integers
Then, for all non-negative compactly supported One can absorb the curvature dependent term in the middle to obtain the following isoperimetric inequality. For its proof see Section 5.2.

Corollary (Isoperimetric inequality). If in addition
and

Preliminaries
In this section, we show how the first variation can be represented by integration, see Lemma 2.1 and Lemma 2.2. These are basic facts that have been frequently used in the study of Euclidean varifolds. We are going to need them in order to prove the Sobolev inequality (Theorem 1.11). Moreover, we prove that any smoothly immersed manifold is a varifold, see Lemma 2.3 and Example 2.4. Finally, we state the Hessian comparison theorems for the distance function (see Lemma 2.5 and Lemma 2.7) that are crucial to derive the monotonicity inequalities in Section 3. Then, there exists a δV measurable map η : N → T N such that δV almost everywhere, g(η, η) ≤ 1, and Remark. Notice that by definition, δV is a Borel regular measure on N . Hence, it is a Radon measure if and only if δV (K) < ∞ for all compact sets K ⊂ N .
where we've used the Einstein summation convention, and ∂ ∂x i α are the coordinate vector fields corresponding to the chart x α . Then, Therefore, we can apply the representation theorem [9, 2.5.12] in combination with Lusin's theorem [9, 2.3.6] to obtain a Borel map k α on U α taking values in the dual space (R n ) * of R n such that δV almost everywhere, k αi k αj g ij α = 1, and For , this means that δV almost everywhere, g(η α , η α ) = 1, and . This equation uniquely determines the Borel map η α up to a set of δV measure zero. Hence, we can define a δV measurable map η : N → T N by requiring that . Choose a finite covering {U α : α ∈ I 0 } of the support of X together with a subordinate partition of unity {ϕ α : α ∈ I 0 }. Then, which finishes the proof.

Lemma
for all compactly supported continuous functions k : M → R. In particular, the push forward measure f # µ f * g of µ f * g under f is a Radon measure on N and satisfies

2)
where H 0 denotes the counting measure. Moreover, for all  tx(b). N (a, b).
is an isometry. This means f # µ f * (i * g) = µ i * g and, by the first case,  {λ : p ∈ U λ } is disjoint. In particular, denoting with χ A the characteristic function of any given set A, Hence, using (2.2) for the special case, Hence, (2.3) follows from the special case by linearity of the limit operator.
To prove (2.4), and there exists 0 < ε < 1 such that the cone where S m−1 := {ξ ∈ T p 1 M : (f * g) p 1 (ξ, ξ) = 1}, as well as the density function By [33, Chapter II, Lemma 5.4], there holds µ Θ * (f * g) = θµ g 0 , where g 0 is the canonical product metric on (0, ε 1 ) × S m−1 . Hence, for E := (Θ • (df p 1 ) −1 )[C] and u 1 := (df p 1 ) −1 (v 1 ), we have by Fubini's theorem where µ f * g E denotes the Radon measure on M given by (µ f * g E)(B) = µ f * g (E ∩B) for all Borel sets B ⊂ M . Hence, by (2. 2) applied to f | U 1 , there holds Θ m Moreover, we make the following observation. Choose ρ > 0 such that B ρ (x) is a geodesic ball around x in N . Given any unit vector v ∈ T x N and δ > 0, we denote with C(v, δ) the image of the set B ρ (0) ∩ {w ∈ T x N : |rw − v| g < δ for some r ∈ R} under the exponential map exp x : B ρ (0) ∩ T x N → N . Then, given any smooth curve γ in N with one can use normal coordinates and differentiability of γ to show that for some t 0 > 0, there holds This observation together with compactness ofB ρ (x) can be used to show that for small In particular, Let X ∈ X (N ). By a simple computation (see [11,Lemma 3 whenever p ∈ M and f (p) = x, where (X • f ) t denotes the orthogonal projection of (X • f ) onto the tangent bundle T M . Integrating this equation and using Lemma 2.3 as well as the usual Divergence Theorem on M (see [33, Chapter II, Theorem 5.11]), we infer where ν is the outward unit normal vector field on ∂M . In particular, V has generalised mean curvature H, for all Borel sets B ⊂ N , and by (2.4), By definition of H, we have trivially

Lemma (See [29, Theorem 6.4.3]). Suppose (N, g) is a Riemannian manifold, p ∈ N , U is a geodesically star-shaped open neighbourhood of p, the metric is represented in geodesic polar coordinates
Then, the Hessian ∇ 2 r of r can be bounded below on U by

Remark. Define the continuous function
where a(0) = 1. Using the series expansion where all higher order terms are negative, we see that a is strictly decreasing. In particular, since cot( π (2.9) (p).
Then, writing r = d(p, ·), there holds Proof. Writing the metric in polar coordinates g = dr ⊗ dr + g r and using Lemma 2.5 in combination with (2.7), we compute for b > 0 Given any T ∈ G m (T U ) with orthonormal basis {e 1 , . . . , e m }, it follows which concludes the proof.

Monotonicity inequalities
In this section, we prove several monotonicity inequalities. The version for 2-dimensional varifolds on general Riemannian manifolds (Lemma 3.1) will be used to prove existence and upper semi-continuity of the density (see Theorem 3.6 in this section), the diameter pinching (see Theorem 1.5 and Section 4.1 for its proof), and part (1) of the Li-Yau inequality (see Theorem 1.7 and Section 3.1 for its proof). The version for 2-dimensional varifolds on non-positively curved manifolds (Lemma 3.7) will be used to prove part (2) of the Li-Yau inequality. The version for general m-dimensional varifolds (Lemma 3.3) is needed to prove the Sobolev inequality (see Theorem 1.11 and Section 5.1 for its proof). The proof of the following Lemma is based on the ideas of the monotonicity formula in Simon [39] in combination with a technique of Anderson [3]. See also [31,Lemma A.3] for a proof in the presence of boundary, and [25] for higher dimensional varifolds.

Lemma. Suppose n is an integer
Then, writing r = d(p, ·), there holds Proof. Given any σ < t < ρ and any non-negative smooth function ϕ : R → R whose support is contained in the interval (−∞, 1), we let X = ϕ( r t )r∇r and compute where ∇ T r denotes the orthogonal projection of ∇r onto T . We write ∇ ⊥ r : and notice that 1 = |∇r| 2 g = |∇ T r| 2 g + |∇ ⊥ r| 2 g . Therefore, testing the first variation equation (see (1.1)) with X, we infer by Lemma 2.7 There holds Hence, adding N 2ϕ( r t )(1 − a b (r)) d V on both sides of the inequality and multiplying with 1 (3.1) Given any 0 < λ < 1, choose ϕ such thatφ ≤ 0, ϕ(s) = 1 for all s ≤ λ, and ϕ(s) = 0 for all s ≥ 1. In other words, ϕ approaches the characteristic function of the open interval (−∞, 1) from below as λ → 1. In particular, ifφ( r t ) = 0, then r t ≥ λ. Hence, Moreover, given any V integrable real valued function f , one computes using Fubini's theorem, writing r σ := max{σ, r}, and denoting with χ A the characteristic function of Therefore, putting (3.2) into (3.1), integrating with respect to t from σ to ρ, letting λ → 1, and using (3.3), we infer where π : G 2 (N ) → N denotes the canonical projection. Observe that Thus, it follows which, in view of (2.7), implies the conclusion.

Remark.
In the above proof, we let ϕ approach the characteristic function of the interval (−∞, 1) from below. If we instead let ϕ approach the characteristic function of the interval (−∞, 1] from above, we obtain the following version for closed balls: for all 0 < σ < ρ. Define the function Then, using the series expansion of cot(x) (see (2.6)), we obtain the series expansion for c: with all higher order terms being positive. In particular, c(0) = 1 3 and c is strictly increasing. Since c( π 2 ) = 4 π 2 , the curvature depending term in Lemma 3.1 can be estimated by Analogously for closed balls: The following lemma is the Riemannian analogue of Corollary 4.5 and Remark 4.6 in [21].
Proof. We proceed similarly as in the proof of Lemma 3.1. Given any σ < t < ρ and any non-negative non-decreasing smooth function ϕ : R → R, we let X = ϕ( r t )r∇r and compute div T X = ϕ r t div T (r∇r) +φ r t r t |∇ T r| 2 g for all T ∈ G 2 (T N ), where ∇ T r denotes the orthogonal projection of ∇r onto T . Writing Adding the term m N (1 − a(r))ϕ( r t ) d V on both sides of the inequality, multiplying by 1 t m+1 , and neglecting positive terms on the right hand side, we infer Integrating the inequality with respect to t from σ to ρ, letting ϕ approach the characteristic function of the interval (−∞, 1] from above and using Lemma 2.1, it follows which implies the conclusion.

Remark. Suppose b > 0 and the function a is given by a(r)
Then,using the series expansion of cot(x) (see (2.6)) we obtain the series expansion for c: with all higher order terms being positive. In particular, c(0) = 0 and c is strictly increasing. Since c( π 2 ) = 2 π , the curvature depending term in Lemma 3.3 can be estimated by The following lemma is a consequence of Lemma 3.3. It can also be derived directly from the first variation formula, see [35,Theorem 5.5]. Proof. Similarly as in (3.3), one can use Fubini's theorem to compute where r σ = max{r, σ}, and r = d(p, ·). For small ρ > 0 we can apply Lemma 2.7 for some b > 0 in combination with Lemma 2.1, Lemma 3.3, and (3.6) to infer Now, only the left hand side depends on σ. Hence, The proof of the following theorem is based on the Euclidean version in the appendix of [18]. See also [32,Corollary 5.8] for the existence of the density.

Theorem. Suppose n is an integer
Then, for all p ∈ N \ spt δV sing , there holds: 1. The density Θ 2 ( V , p) exists and is finite.
Proof. Given any non-negative smooth function ϕ : R → R whose support is contained in the open interval (−∞, ρ 0 ), we define the vector field X := ϕ(r)s b (r)∇r. Write the metric in polar coordinates g = dr ⊗ dr + g r on U and use Lemma 2.5 to estimate Hence, which implies div T X ≥ ϕ ′ (r)s b (r)|∇ T r| 2 g + 2ϕ(r)c b (r) for all T ∈ G 2 (T U ), where ∇ T r denotes the orthogonal projection of ∇r onto T . Writing and testing the first variation equation (see (1.1)) with X, we infer (3.13) Notice that the function t → V B t (p) is continuous at t 0 if and only if V ({r = t 0 }) = 0. Since the function t → V B t (p) is non-decreasing, it can only have countably many discontinuity points. Choose 0 < σ < ρ < ρ 0 to be continuity points. Define the non-increasing Lipschitz function and let ϕ approach (φ σ (·) − φ(ρ)) + , where (·) + := max{·, 0}. Then, by the dominated convergence theorem, (3.13) becomes where π : G 2 (N ) → N is the canonical projection. We compute Putting (3.15) and (3.16) into (3.14) and neglecting negative terms on the right hand side implies the conclusion.

Proof of Theorem 1.7
If b ≥ 0, the theorem is a consequence of Lemma 3.1 in combination with (3.5). Indeed, by Theorem 3.6, we can multiply the inequality with 4, let σ → 0+ and let ρ → ∞ to conclude the positive curvature case. Now suppose that b < 0. We are going to determine the limits in Lemma 3.7 as σ → 0+ and ρ → ∞. Using L'Hôspital's rule twice, one readily verifies Therefore, by Theorem 3.6, Similarly, by L'Hôspital's rule, Hence, by Hölder's inequality and square integrability of the generalised mean curvature, All the other limits can be easily determined using that spt V is compact and using that p / ∈ spt δV sing .

Diameter bounds
In this Section, we provide the lower diameter bounds for varifolds (Lemma 4.1 and Lemma 4.3) that are needed to prove Theorem 1.2, Theorem 1.4, and Theorem 1.5. To prove Lemma 4.3 we will need the Hessian comparison theorem of the distance function for asymptotically non-positively curved manifolds (Lemma 4.2). At the end of this section, we will prove Theorem 1.5.
The following lemma is a direct combination of the representation formula for the first variation, Lemma 2.1, with Rauch's comparison theorem, Lemma 2.7. Then, the Hessian ∇ 2 r of r can be bounded below by Proof. Using Lemma 4.2, we compute ∇(r∇r) = dr ⊗ dr + r∇ 2 r and thus div T (r∇r) ≥ m 1 + √ 1 − 4b 2 for all T ∈ G m (T D(p)). Now we can test the first variation formula (1.1) with the vector field r∇r and use Lemma 2.1 to conclude the proof.

Proof of Theorem 1.5
The following proof is based on Simon [39, Lemma 1.1].
Let h ≤ 1 be a non-negative compactly supported C 1 function on N and define the varifold V h ∈ V m (N ) by letting for all compactly supported continuous functions k on G m (N ). Given any X ∈ X (N ), we compute for T ∈ G m (T N ) (see [11,  where π : G m (N ) → N is the canonical projection. With the usual approximation from above (see [9, 2.1.3 (5)]), one can see that the inequality remains valid for U replaced with any closed set. In particular, δV h is a Radon measure. Hence, we can apply Lemma 3.3 in combination with Lemma 2.1, Lemma 2.7, and (3.7) to deduce for all p ∈ N and 0 < σ < ρ < min{i(spt V ), π Given any p ∈ spt V with Θ * m ( V h , p) ≥ 1, Lemma 5.1 applied with f (t) = α(m) −1 V h B t (p) and implies by (5.1) |∇ ⊤ h| g dV . Now, Vitali's covering theorem (see [9, 2.8.5, 6, 8]) implies the assertion.

Proof of Corollary 1.12
First, we apply Theorem 1.11 with h approaching the constant function 1 from below, to deduce V (N ) ≤ C (5.2) Applying this inequality again on the right hand side, we inductively infer for all positive integers k ≥ 1,