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

Arch. Math. 115 (2020), 215–228 c 2020 The Author(s)  0003-889X/20/020215-14 published online May 8, 2020 https://doi.org/10.1007/s00013-020-01461-4 Archiv der Mathematik Enclosure and non-existence theorems for area stationary currents and currents with mean curvature vector Patrick Henkemeyer Abstract. 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. Mathematics Subject Classification. 49Q15, 49Q20, 53A10, 58A25. Keywords. Enclosure theorems, Non-existence theorems, Area stationary currents, Currents with mean curvature vectors. 1. Introduction. Let Ω ⊂ R2 denote a domain and let X : Ω → R3 be a minimal surface, i.e. a harmonic and conformal mapping of class C 2 . For detailed information on minimal surfaces, we refer to the monographs of Dierkes, Hildebrandt, Sauvigny, and Tromba [4] and [5]. It is well-known that X(Ω̄) is contained in the convex hull of its boundary components X(∂Ω). This result holds true for every harmonic mapping due to the maximum principle. Together with the conformality conditions, Hildebrandt [10] obtained stronger results. In fact, he proved that a minimal surface X(Ω̄) is enclosed by the hyperboloid H(R) := {(x, y, z) ∈ R3 : x2 + y 2 − z 2 ≤ R} for all R ∈ R provided X(∂Ω) ⊂ H(R). For R = 0, one gets a cone and the theorem still holds true. Furthermore one has that the minimal surface cannot pass through the vertex of the cone. Hildebrandt [10] also stated analogous results for two dimensional H-surfaces. Again, by using the maximum principle for elliptic equations, Dierkes [3] and Dierkes and Schwab [6] extended these results to compact n-dimensional C 2 -submanifolds M in Rn+k . By regularity, these results immediately lead P. Henkemeyer 216 Arch. Math. 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 Rn+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 ⊂ Rn+k be an open set. We write T = τ (M, θ, ξ) ∈ Rn (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 μT := Hn θ and δT  is the total variation measure. A current with mean curvature H = −DμT δT σ is given by Definition 1.1. Let T = τ (M, θ, ξ) ∈ Rn (U ) be a current and H ∈ L1loc (M ∩ U, Rn+k ; μT ). Then T has mean curvature vector H in U if   divM X dμT = − H, X dμT (1.1) U whenever X ∈ U Cc1 (U \ spt ∂T, Rn+k ). An important special case is H(x) ≡ 0 leading to the definition of “stationary currents” . 2. Stationary currents. 2.1. Enclosure theorem. The following convex hull property for stationary currents is well-known, cf. [15, Thm. 19.2, Rmk. 34.2]. Theorem 2.1. Let T ∈ Rn (Rn+k ) be a stationary current in Rn+k with compact support. Then spt T ⊂ conv(spt ∂T ). We prove an enclosure result in non-convex sets. Therefore we define for n+k−j n+k j = 1, . . . , n − 1, rj (x) := i=1 x2i and sj (x) := i=n+k−j+1 x2i , and the quadratic function qj (x) = qj (x1 , . . . , xn+k ) : Rn+k → R given by qj (x) := n+k rj (x) − n−j : qj (x) ≤ R} be a j sj (x). Furthermore let Hj (R) := {x ∈ R generalized hyperboloid for some R ∈ R. Vol. 115 (2020) Enclosure and non-existence theorems 217 Theorem 2.2 (Enclosure theorem). Let Hj (R) be a generalized hyperboloid for R ∈ R and j = 1, . . . , n − 1 congruent to Hj (R). Let T ∈ Rn (Rn+k ) be a stationary current in Rn+k with compact support spt T and let the boundary values satisfy spt ∂T ⊂ Hj (R). Then we have spt T ⊂ Hj (R). Proof. W.l.o.g. assume Hj (R) = Hj (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, γ  (t) > 0 for t > R + ε. Define x̂ : Rn+k → Rn+k by   n−j n−j xn+k−j+1 , . . . , − xn+k x̂(x) := x1 , . . . , xn+k−j , − j j and consider X(x) := Ψspt T (x) γ(qj (x)) x̂(x) ∈ Cc1 (Rn+k \Hj (R), Rn+k ), where Ψspt T (x) ≡ 1 in a neighborhood of spt T is a smooth cut-off function. Let Tx M denote the approximate tangent space of T in x ∈ M (which exists Hn -a.e.) and PTx M : Rn+k → Tx M the orthogonal projection with matrix representation (pij )i,j=1,...,n+k w.r.t. the canonical basis of Rn+k . We often abbreviate the projection by ( · ) . For this vector field X, we have divM X = ∇M γ, x̂ + γ divM x̂ and we calculate the different expressions. Firstly, ∇M γ(qj ) = γ  (qj ) (Dqj ) = 2γ  (qj ) x̂ which gives ∇M γ(qj ), x̂ = 2γ  (qj ) x̂ , x̂ = 2γ  (qj ) |x̂ |2 ≥ 0 and secondly, divM x̂ = n+k  ∇M x̂i , ei i=1 = n+k−j  ∇M xi , ei − i=1 = n+k  = n− n+k  pii − i=1 n−j j i=n+k−j+1 n j n+k  n+k  ∇M xi , ei i=n+k−j+1 n−j pii − j n+k  pii i=n+k−j+1 pii ≥ 0. i=n+k−j+1 We have used tr(PTx M ) = n in the last equation. Plugging these into (1.1) yields  0=  n γ(qj ) n − j Rn+k n+k   pii + 2 γ  (qj ) |x̂ |2 dμT . i=n+k−j+1 For all i = n + k − j + 1, . . . , n + k, we have pii = 1 iff ei ∈ Tx M . Thus for μT -a.e. x ∈ M ,  n n− j n+k  i=n+k−j+1  pii = 0 iff pii = 1 for all i = n + k − j + 1, . . . , n + k. P. Henkemeyer 218 Arch. Math. Otherwise it is strictly positive. Define the set E := {x ∈ M : Tx M exists and en+k−j+1 , . . . , en+k ∈ Tx M }. We have on the one hand,    n+k  n γ(qj ) n − pii dμT = 0 j Rn+k \E i=n+k−j+1 where the expression {. . .} is positive and due to the definition of γ, this means spt μT ∩ (Rn+k \E) ⊂ {x ∈ Rn+k : qj (x) ≤ R + ε} = Hj (R + ε). On the other hand, notice that {. . .} = 0 in E, thus     n+k  n pii + γ  (qj ) |x̂ |2 dμT = γ  (qj ) |x̂ |2 dμT . 0 = γ(qj ) n − j E i=n+k−j+1 E  2 Now notice that |x̂ | = 0 iff x̂ ⊥ T (...truncated)