Simultaneous stability of $C^{\infty }$ convex integrands and their duals

Abstract

In this paper, the following three are shown. (1) For a $C^{\infty }$ convex integrand $\gamma :S^n\to {\mathbb{R}}_{+}$, its dual convex integrand $\delta :S^n\to {\mathbb{R}}_{+}$ is of class $C^{\infty }$. (2) For a stable convex integrand $\gamma :S^n\to {\mathbb{R}}_{+}$, its dual convex integrand $\delta :S^n\to {\mathbb{R}}_{+}$ is stable. (3) Let $\gamma :S^n\to {\mathbb{R}}_{+}$ be a stable convex integrand. Then, for any $i$ $(0\le i\le n)$, ${\theta }_0\in S^n$ is a non-degenerate critical point of $\gamma$ with Morse index $i$ if and only if $-{\theta }_0\in S^n$ is a non-degenerate critical point of the dual convex integrand $\delta$ with Morse index $(n-i)$.

This paper has been withdrawn by the author due to important improvements to be done