Hot spots on cones and warped product manifolds

2025/08/25 by Hatcher, Lawford
#35B38 #35J05 #35J25 #35P05 #58J50 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.18054

Abstract

We study extrema of solutions to the heat equation (i.e. hot spots) on certain types of warped product manifolds of the form ([0,L]× M,dr²+f(r)²h) where (M,h) is a closed Riemannian manifold. We prove that, under certain conditions on the warping function f, the statement of Rauch's hot spots conjecture holds for the corresponding warped product. We then go on to study the long-time behavior of hot spots on infinite cones over closed Riemannian manifolds. In this case, under appropriate hypotheses on the initial condition, there are four possible long-time behaviors depending only on the spectral gap of the fiber (M,h).

Citations

• Hot spots in domains of constant curvature
• On the hot spots conjecture in higher dimensions
• A hot spots theorem for the mixed eigenvalue problem with small Dirichlet region
• The hot spots conjecture for some non-convex polygons
• On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions
• Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles
• First mixed Laplace eigenfunctions with no hot spots
• Heat kernel estimate in a conical singular space
• Closed nodal lines and interior hot spots of the second eigenfunction of the Laplacian on surfaces

Related

• The spectrum of the Laplacian on closed manifolds and the heat asymptotics near conical points
• The hot spots conjecture on Riemannian manifolds with isothermal coordinates
• Heat Kernel on Warped Products
• Hot spots of solutions to the heat equation with inverse square potential
• Blowing-up solutions concentrating along minimal submanifolds for some\n supercritical elliptic problems on Riemannian manifolds