A new proof of a characterization of small spherical caps

TL;DR

This paper offers a novel proof for the characterization of small spherical caps as the only constant mean curvature graphs bounded by a circle, using flux formulas and integral equalities instead of the traditional maximum principle.

Contribution

It introduces a new proof method based on flux formulas and integral equalities, differing from the classical maximum principle approach.

Findings

Characterization of small spherical caps as unique constant mean curvature graphs bounded by a circle.

Development of a proof utilizing flux formulas and integral equalities.

Provides an alternative proof inspired by Reilly's work.

Abstract

It is known that planar disks and small spherical caps are the only constant mean curvature graphs whose boundary is a round circle. Usually, the proof invokes the Maximum Principle for elliptic equations. This paper presents a new proof of this result motivated by an article due to Reilly. Our proof utilizes a flux formula for surfaces with constant mean curvature together with integral equalities on the surface.