On the first Neumann eigenvalue for critical points of a weighted area functional with asymptotically flat ends

2025/07/28 by Martínez-Triviño, A. L.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.20955

Abstract

In the following work, we obtain a lower bound for the first Neumann eingevalue of the drift Laplacian Δ^φ for a family of properly embedded [φ,e₃]-minimal surfaces in ℝ³ with concave function φ and asymptotically flat ends. As application, we obtain a control on the topology for that surfaces with finite total curvature. The strategy will consists in an integration of the Bouchner's formula by the works of A. Lichnerowicz and S. Brendle, R. Tsiamis and relate said eigenvalue with the Poincare's constant.

Citations

• Stability of vertical and radial graphs in the Euclidean space with density
• On eigenfunctions and nodal sets of the Witten-Laplacian
• A Weierstrass type representation for translating solitons and singular minimal surfaces
• What is the Shape of a Cupola?
• Equilibrium of Surfaces in a Vertical Force Field
• A Calabi's Type Correspondence
• Graphical Translators for Mean Curvature Flow
• Simons' type equation for f-minimal hypersurfaces and applications
• Rigidity of gradient Ricci Solitons
• Convex solutions to the mean curvature flow
• Ricci flow with surgery on three-manifolds
• The entropy formula for the Ricci flow and its geometric applications

Related

• First eigenvalue of embedded minimal surfaces in S³
• Inequalities between Dirichlet and Neumann Eigenvalues on Surfaces
• Maximization of the first nontrivial eigenvalue on the surface of genus two
• Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds
• Large topology asymptotics for spectrally extremal minimal surfaces in \mathbbB³ and \mathbbS³