The hyperplane is the only stable, smooth solution to the Isoperimetric Problem in Gaussian space

Abstract

Abstract. We study stable, two-sided, smooth, properly immersed solutions to the Gaussian Isoperimetric Problem. That is, we study hyper-surfaces Σ n ⊂ R n+1 that are second order stable critical points of minimizing Aµ(Σ) = Σ e −|x| 2 /4 dA for compact variations that preserve weighted volume. Suchshow that such Σ satisfy the curvature condition H = x, N /2 + C where C is a constant. We also derive the Jacobi operator L for the second variation of such Σ. Our first main result is that for non-planar Σ, bounds on the index of L, acting on volume preserving variations, gives us that Σ splits off a linear space. A corollary of this result is that hyper-planes are the only stable smooth, complete, properly immersed solutions to the Gaussian Isoperimetric Problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of Σ 2 ⊂ R 3 , there is a gradient decay estimate for fixed bound |C| ≤ M (C is from the curvature condition) and Σ obeying an appropriate Aµ condition. This shows that for fixed C, in the limit as R → ∞, stable (Σ, ∂Σ) ⊂ (B 2R (0), ∂B 2R (0)) with good volume growth bounds approach hyper-planes.

Volume Growth of Complete Submanifolds in Gradient Ricci Solitons with Bounded Weighted Mean Curvature

“…In [2] and [3], Barbosa-do Carmo-Eschenburg showed that a constant mean curvature (CMC) hypersurface in a Riemannian manifold is the critical point of its area functional for compactly supported variations which preserve enclosed volume. Mcgonagle-Ross [24] proved the same property for CWMC hypersurfaces in R m under the Gaussian weighted area and Gaussian weighted volume, respectively. Using the similar argument to the one in [3] and [24], one can show that a CWMC hypersurface in a Riemannian manifold is still the critical point of its weighted area functional for compactly supported variations which preserve enclosed weighted volume, where the weighted volume element in M is e −f dv M and dv M denotes the volume element of M .…”

“…So an f -minimal hypersurface Σ is the case of C = 0, that is, H = ∇f, n . When (M, g) is the Euclidean space (R m , g 0 ) with the Gaussian measure e − |x| 2 4 dv g 0 , a CWMC hypersurface is just the solution to the Gaussian isoperimetric problem in [24] or the λ-hypersurface in [6] (observe that in [6], the weighted measure is taken to be e − |x| 2 2 dv g 0 , which has no essential difference).…”

“…A trivial example is any hyperplane as H = 0 and x, n is constant. In [24], they proved that hyperplanes are the only stable smooth, complete, properly immersed solutions to the Gaussian isoperimetric problem, and that there are no hypersurfaces of index one. Later, in [6], Q.M.Cheng-G.Wei called hypersurfaces satisfying (1.1) λ-hypersurfaces.…”

Variational characterizations of $$\xi $$-submanifolds in the Eulicdean space $${{\mathbb {R}}}^{m+p}$$

“…Problem: Naturally we believe and expect that the additional condition (1.4) in Theorem 7.1 could be dropped; Furthermore, motivated by the main theorem of [27], it is also expected, without any additional conditions, that the m-planes are the only properly immersed, complete W -stable ξ-submanilds or, if it is not the case, more examples could be found. Remark 1.1.…”

A new pinching theorem for complete self-shrinkers and its generalization

“…More generally, we consider the rigidity of λ-hypersurfaces. The concept of λhypersurfaces was introduced independently by Cheng-Wei [7] via the weighted volume-preserving mean curvature flow and McGonagle-Ross [25] via isoperimetric type problem in a Gaussian weighted Euclidean space. Precisely, the hypersurfaces of Euclidean space satisfying the following equation are called λ-hypersurfaces:…”