Enclosure and non-existence theorems for area stationary currents and currents with mean curvature vector

We discuss certain geometric properties for area stationary currents and currents with integrable mean curvature, so called “enclosure theorems”. As a consequence, we obtain non-existence results for currents with connected support. Finally, we extend these results to currents in submanifolds and state a non-existence result for stationary currents in spheres.

to non-existence theorems. In fact, there are no smooth connected minimal submanifolds with boundary components in both disjoint parts of a special cone.
The question arises whether this cone can be enlarged or not. For two dimensional minimal surfaces, this was answered by Osserman and Schiffer [13]. They give the optimal "non-existence" cone. Dierkes [3] proved the corresponding theorem for n-dimensional smooth submanifolds in R n+1 . See also the monograph Dierkes et al. [5,Ch. 4] for a complete survey of these results.
We mention that there are more general results of this type in various situations, cf. [1,9,19], and [2].
Here we want to address the following question: Do these theorems also generalize to currents? We show in the sequel that this is basically the case. The classical maximum principle will be replaced by a maximum principle of Solomon and White [16]. Maximum principles for singular surfaces in general situations were studied by different authors in the last years. We mention [11,14], and recently [18] for codimension 1 and [17] where certain varieties of arbitrary codimension are considered.
We use the notation of Simon [15]. Let n ≥ 2, k ≥ 1 be natural numbers and U ⊂ R n+k be an open set. We write T = τ (M, θ, ξ) ∈ R n (U ) for the set of n-dimensional rectifiable currents in U . Our notation slightly differs from most authors as we do not require integer multiplicity θ. As usual, we define the associated Radon measure μ T := H n θ and δT is the total variation measure. A current with mean curvature H = −D μT δT σ is given by whenever X ∈ C 1 c (U \ spt ∂T, R n+k ).
An important special case is H(x) ≡ 0 leading to the definition of "stationary currents" . We prove an enclosure result in non-convex sets. Therefore we define for j = 1, . . . , n − 1, r j (x) := x 2 i and s j (x) := n+k i=n+k−j+1 x 2 i , and the quadratic function q j (x) = q j (x 1 , . . . , x n+k ): R n+k → R given by q j (x) := r j (x) − n−j j s j (x). Furthermore let H j (R) := {x ∈ R n+k : q j (x) ≤ R} be a generalized hyperboloid for some R ∈ R. Proof. W.l.o.g. assume H j (R) = H j (R). Let ε > 0 be arbitrary and γ ∈ C 1 (R) be non-negative and non-decreasing with γ(t) ≡ 0, t ≤ R + ε, and γ(t) > 0, Let T x M denote the approximate tangent space of T in x ∈ M (which exists H n -a.e.) and P TxM : R n+k → T x M the orthogonal projection with matrix representation (p ij ) i,j=1,...,n+k w.r.t. the canonical basis of R n+k . We often abbreviate the projection by ( · ) .
For this vector field X, we have div M X = ∇ M γ,x + γ div Mx and we calculate the different expressions. Firstly, We have used tr(P TxM ) = n in the last equation. Plugging these into (1.1) yields For all i = n + k − j + 1, . . . , n + k, we have p ii = 1 iff e i ∈ T x M . Thus for μ T -a.e. x ∈ M , Otherwise it is strictly positive. Define the set E := {x ∈ M : T x M exists and e n+k−j+1 , . . . , e n+k ∈ T x M }. We have on the one hand, where the expression {. . .} is positive and due to the definition of γ, this means In other words, if x n+k−j+l = 0 for (at least) some l = 1, . . . , j, then |x | 2 > 0 in E. Again, because of γ, this establishes and combining both results, we arrive at . , x n+k = 0} and is strict positive anywhere else in some sufficiently large ball. Then, sincẽ x is off the closed hyperboloid together with the fact (2.1), we have that in a small neighborhood points of spt T fulfill x n+k−j+1 = 0, . . . , x n+k = 0. By definition of f , this means in particular Notice that, because ofx ∈ E, we have e n+k−j+1 , . . . , e n+k ∈ TxM . In other words, This leads to a contradiction to the definition of the approximate tangent space θ(x) TxM f (y) dH n (y) > 0. This gives spt T = spt μ T ⊂ H j (R + ε) and since ε > 0 was arbitrary, the theorem is established.

Non-existence theorem.
For j = 1 and R = 0, we get the cone Vol. 115 (2020) Enclosure and non-existence theorems 219 Lemma 2.4. Let T ∈ R n (R n+k ) be a stationary current in R n+k and spt T ⊂ K. Then both currents T + ∈ R n (R n+k ) and T − ∈ R n (R n+k ) are also stationary in R n+k .
Proof. We consider T + . Let ε > 0 be arbitrary and φ ε (t) ∈ C 1 (R) be non- for some (other) c > 0, which depends only on the angle of the cone, i.e. on the dimension n. From monotonicity for stationary currents, the estimate μ T (B cε (0)) ≤ cε n holds true for small ε > 0. Let X be an arbitrary compactly supported C 1 vector field in R n+k and spt X ∩ spt ∂T + = ∅. In view of the stationarity of T , we can test φ ε (x n+k )X. Thus The same inequality holds true for −X and establishes the stationarity of T + .
Applying Theorem 2.1 to T + and T − , respectively, together with the fact that spt ∂T ± ⊂ K ± , we get 0 ∈ spt T . Since all calculations are invariant under translations, we have proved the following theorem.
Then there exists no stationary current T ∈ R n (R n+k ) such that spt T is compact and connected with boundary values spt ∂T ⊂ K + ∪ K − such that both spt ∂T ∩ K + and spt ∂T ∩ K − are non-empty.
Remark. As n increases, the angle of aperture β of K is increasing as well. Precisely we have β = arctan( √ n − 1) and therefore β → 90 • as n → ∞. The enclosure and non-existence theorems extend the existence theorem for currents with integer multiplicity, cf. [15,Lem. 34.1], as we get more information of the solution, e.g., the shape or the disconnectedness in at least two parts.
The proof of the following is the same as in [

Optimal results in codimension one.
The question arises if it is possible to "enlarge" the cone and still prove non-existence. We restrict ourselves to codimension k = 1. We use Dierkes' [3] construction of n-dimensional catenoids enclosing a cone with a larger angle of aperture. Therefore we show that the enclosing procedure also holds true for currents.
Consider a curve (x, y(x)) in R 2 and its rotational symmetric n-dimensional , ω ∈ S n−1 }. Its area is proportional to the one dimensional integral Proposition 2.7 (Enclosure result.) Let T ∈ R n (R n+1 ) be a stationary current in R n+1 with compact support and spt ∂T ⊂ K = {x ∈ R n+1 : ρ < ±τ 0 x n+1 }. Then we have spt T ⊂K.
Proof. We will show spt T ⊂ C a for all a > 0. Because of a>0 C a =K, the proposition is then established. Let us assume spt T ⊂ C a for some a > 0 and consider η 0,λ# T ∈ R n (R n+1 ) with minimal λ > 1 s.t. spt(η 0,λ# T ) ⊂ C a . Then there exists p ∈ spt(η 0,λ# T ) with p ∈ ∂C a . The contracted current lies completely on one side of the smooth submanifold ∂C a and touches it at least Vol. 115 (2020) Enclosure and non-existence theorems 221 in p. We want to apply the maximum principle of Solomon and White [16]. Therefore let us abbreviate for the fixed λ: η(x) := η 0,λ (x) andT := η 0,λ# T = η # T . Claims: in η(x) and we have equality. (iii) Let T be stationary in R n+1 , then so isT = τ (η(M ), θT , ξT ).
Claim (i) and (ii) are proved by direct calculations. For the third statement, note that we have for the Jacobian, J M η = λ −n , cf. [15,Ch. 12 We calculate the variation of the contracted current We have for the first factor where we have used the second claim. Finally, we get by applying the area formula ThusT is stationary in R n+1 because Y was arbitrary. Now we are able to apply the maximum principle [16] since the support of T is compact. Therefore 222 P. Henkemeyer Arch. Math.
we get coincidence of sptT and ∂C a in an open subset. This is a contradiction to claim (i).
With exactly the same argument as in Section 2.2, we can prove the nonexistence theorem. Furthermore it is optimal because the cone is enclosed by a family of minimal submanifolds.
Then there is no stationary current T ∈ R n (R n+1 ) with compact support such that spt T is connected and the boundary fulfills spt ∂T ⊂ K + ∪K − such that spt ∂T ∩K + as well as spt ∂T ∩K − is non-empty.

Currents with mean curvature vector.
3.1. Enclosure theorem. The enclosure Theorem 2.2 naturally extends to currents with mean curvature vector H ∈ L 1 loc (μ T ) under appropriate conditions. To this end, we define the quadratic function q j (x) with an additional param- Proof. The proof is similar to the case H ≡ 0. Definê . Note that the additional term can be estimated directly with the Cauchy-Schwarz inequality H(x), . Plugging all terms into (1.1) gives Vol. 115 (2020) Enclosure and non-existence theorems 223 We can estimate the expression {. . .} pointwise and now notice that the resulting term is non-negative in R n+k \H j (R) for μ T -a.e. x. Next we integrate over the sets E and R n+k \E where E is defined exactly as above. In the latter case, we evidently have strict inequality in (3.2) as not all p ii are equal to one. Thus spt μ T ∩ (R n+k \E) ⊂ H j (R). The proof is now finished as before.
We mention two sufficient characterizations for the mean curvature vector H such that the condition (3.1) is satisfied. Both are easier to check.
In fact, we have as in [15].

Proposition 3.3.
Let V be a rectifiable varifold in some neighborhood of the origin. Suppose the density exists in 0 and for V j := η 0,λj # V, λ j 0, we have μ Vj → μ C in R n+k such that μ C is associated to a rectifiable varifold C which is stationary in R n+k . Then C is a cone. Proof. We show x ∈ A implies the existence of a ρ > 0 such that μ(B ρ (x)) = 0. From the hypothesis, we have dist(x, A) > ε > 0 and therefore define ρ := 1 3 dist(x, A). Notice, because of spt μ j ⊂ A for all j ∈ N, it evidently follows B 2ρ (x) ∩ spt μ j = ∅ for all j ∈ N. Now choose a continuous function φ : this completes the proof.
We consider the two restricted varifolds This gives spt C ± ⊂ {x n+k = 0} in contradiction to spt C ± ⊂K = K because by construction evidently C ≡ 0. Notice in particular this means that the support of a stationary tangent cone cannot pass through the vertex of the cone K and therefore the origin cannot be contained in the support of the current T . In order to fulfill the assumptions of Proposition 3.3, we assume H ∈ L p loc (U, R n+k ; μ T ) for some p > n and θ ≥ 1 μ T -a.e. in U for some small neighborhood U of the origin with U ∩ spt ∂T = ∅. This gives the existence of an area stationary tangent cone in every x ∈ spt T ∩ U , see [15,Ch. 8]. Finally, this establishes Theorem 3.7 (Non-existence theorem). Let T ∈ R n (R n+k ) be a current with compact support spt T and mean curvature vector H. Assume for some b ∈ [0, 1], spt ∂T ⊂ K ± := x ∈ R n+k : Vol. 115 (2020) Enclosure and non-existence theorems 225 such that both spt ∂T ∩ K + and spt ∂T ∩ K − are non-empty. Furthermore define r 1 (x) := x 2 1 + · · · + x 2 n+k−1 and suppose the mean curvature vector H satisfies on the one hand and on the other hand, in some neighborhood U of the origin, H ∈ L p loc (U, R n+k ; μ T ) for some p > n and θ(x) ≥ 1 for μ T -a.e. x ∈ U . Then spt T cannot be connected.
Remark. Instead of condition (3.3), one of the requirements of Lemma 3.2 with j = 1 and R = 0 can also be fulfilled. The condition θ ≥ 1 μ T -a.e. is automatically given for currents with integer multiplicity.

Enclosure and non-existence theorems for currents in submanifolds.
Here we discuss an important modification of stationarity in Euclidean spaces. More generally we let N be an (n + l)-dimensional C 2 -submanifold of R n+k for 0 ≤ l ≤ k. We denote by B y : T y N × T y N → (T y N ) ⊥ the second fundamental form of N at y. Then T is called stationary in N if the first variational formula holds true with H M :=   If spt ∂T ⊂ H j (R), then spt T ⊂ H j (R). Let (4.1) be true for j = 1 and R = 0 and assume for ε > 0, on the one hand, θ ≥ 1 μ T -a.e. in B ε (0) and on the other hand, H M ∈ L p loc (B ε (0), R n+k ; μ T ) for some p > n. Furthermore let spt ∂T ⊂ K ± such that both spt ∂T ∩ K + = ∅ and spt ∂T ∩ K − = ∅. Then spt T cannot be a connected set.
We want to estimate the abstract curvature H M against some quantity depending only on the submanifold N . We restrict ourselves to the case l = k − 1. As usual, we can define the principal curvatures κ 1 , . . . , κ n+k−1 of N which we order with respect to their absolute value: |κ 1 | ≥ · · · ≥ |κ n+k−1 |. We define Λ n := |κ 1 | + · · · + |κ n | what we call the n-mean curvature of N . This number slightly differs from the usual n-mean curvature used in barrier principles of higher codimension, cf. [6,7]. We are now able to proceed similarly as in [12] and arrive at the important estimate: For μ T -a.e. x ∈ M , we have |H M (x)| = tr(B x )| TxM ≤ Λ n (x). This is just a conclusion of the following general result where we skip the proof.