Shape stability of a quadrature surface problem in infinite Riemannian manifolds

TL;DR

This paper investigates the shape stability of quadrature surfaces within infinite-dimensional Riemannian manifolds, emphasizing the role of mean curvature and covariant derivatives in characterizing optimal shapes.

Contribution

It introduces a novel control method for optimal shape characterization using the geometry of infinite-dimensional Riemannian manifolds and covariant derivatives.

Findings

Shape stability depends solely on mean curvature at critical points.

The covariant derivative is crucial for analyzing shape Hessian properties.

The framework applies to infinite-dimensional Riemannian manifolds.

Abstract

In this paper, we give a simple control on how an optimal shape can be characterized. The framework of Riemannian manifold of infinite dimension is essential. And the covariant derivative plays a key role in the computation and in the analysis of qualitative properties from the shape hessian. The control depends only on the mean curvature of the domain which is a minimum or a critical point.