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Serrin's type Problems in convex cones in Riemannian manifolds

Murilo Araújo, Allan Freitas, Márcio Santos, Joyce Sindeaux

TL;DR

This work extends Serrin-type overdetermined problems to cones in warped product and Euclidean settings. By combining a Pohozaev-type identity with a $P$-function approach and exploiting closed conformal vector fields, it proves rigidity: sector-like domains in convex cones must be metric balls $\Sigma \cap B_r(x_0)$ under natural curvature and compatibility conditions, with explicit radial solutions in model spaces. It also derives soap-bubble-type and Heintze-Karcher inequalities in warped-cone contexts, linking boundary mean curvature, volume, and domain shape. Finally, it analyzes a weighted (drift) Serrin problem in Euclidean cones, obtaining analogous rigidity results for the drift Laplacian with homogeneous weights, including an explicit radial solution $u = r^2 - \frac{|x|^2}{2(n+\alpha)}$, thereby unifying geometric-analytic methods across unweighted and weighted geometries.

Abstract

In this work, we discuss several results concerning Serrin's problem in convex cones in Riemannian manifolds. First, we present a rigidity result for an overdetermined problem in a class of warped products with Ricci curvature bounded below. As a consequence, we obtain a rigidity result for Einstein warped products. Next, we derive a soap bubble result and a Heintze-Karcher inequality that characterize the intersection of geodesic balls with cones in these spaces. Finally, we analyze the analogous ovedetermined problem for the drift Laplacian, where the ambient space is a cone in the Euclidean space.

Serrin's type Problems in convex cones in Riemannian manifolds

TL;DR

This work extends Serrin-type overdetermined problems to cones in warped product and Euclidean settings. By combining a Pohozaev-type identity with a -function approach and exploiting closed conformal vector fields, it proves rigidity: sector-like domains in convex cones must be metric balls under natural curvature and compatibility conditions, with explicit radial solutions in model spaces. It also derives soap-bubble-type and Heintze-Karcher inequalities in warped-cone contexts, linking boundary mean curvature, volume, and domain shape. Finally, it analyzes a weighted (drift) Serrin problem in Euclidean cones, obtaining analogous rigidity results for the drift Laplacian with homogeneous weights, including an explicit radial solution , thereby unifying geometric-analytic methods across unweighted and weighted geometries.

Abstract

In this work, we discuss several results concerning Serrin's problem in convex cones in Riemannian manifolds. First, we present a rigidity result for an overdetermined problem in a class of warped products with Ricci curvature bounded below. As a consequence, we obtain a rigidity result for Einstein warped products. Next, we derive a soap bubble result and a Heintze-Karcher inequality that characterize the intersection of geodesic balls with cones in these spaces. Finally, we analyze the analogous ovedetermined problem for the drift Laplacian, where the ambient space is a cone in the Euclidean space.
Paper Structure (4 sections, 15 theorems, 93 equations, 1 figure)

This paper contains 4 sections, 15 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rigidity for a Serrin's type problem in convex cones
  3. Soap Bubble and Heintze-Karcher inequality in cones
  4. The weighted Serrin's problem for convex cones of the Euclidean space

Key Result

Theorem A

Let $M = I \times_{\rho} N$ be a warped product manifold, where $N$ is an $(n-1)$-dimensional Riemannian manifold, $I= [0,R)$ is an interval, and $\rho' > 0$ in $I$. Let $\Sigma$ be a convex cone in $M$ such that $\Sigma \setminus \{O\}$ is smooth, and let $\Omega \subset \Sigma$ be a sector-like do to the problem SP_cone1. If $\operatorname{Ric} \geq (n-1)k g$ and then $\Omega = \Sigma \cap B_{r

Figures (1)

  • Figure 1: A sector-like domain $\Omega$ inside $\Sigma$

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem A
  • Corollary 1.2
  • Theorem B
  • Theorem C
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 12 more