2013/07/26 by Matthew McGonagle, McGonagle, Matthew, John Ross +1 · 2 citations
Mathematics · #53A10 (Primary) 49Q10 #53C42 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #math.DG #msc:49Q10 #msc:53A10 #msc:53C42
paper · pdf · doi:10.48550/arxiv.1307.7088
20 pages
openalex publication_date 2013/07/26 · arxiv created 2014/12/08 · arxiv updated 2014/12/10 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces Σn ⊂ \mathbb Rn+1 that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We show that such Σ satisfy a curvature condition, and 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 hyperplanes are the only stable smooth complete solutions to this Gaussian isoperimetric type problem, and that there are no hypersurfaces of index one. Finally, we show that for the case of Σ2 ⊂ \mathbb R3, there is a gradient decay estimate depending on bounds for the curvature condition and an appropriate area growth bound. This shows that, in the limit as R → ∞, stable (Σ, ∂Σ) ⊂ (B2R(0), ∂ B2R(0)) with good area growth bounds approach hyperplanes.