Isoperimetry and volume preserving stability in real projective spaces

Abstract

We classify the volume preserving stable hypersurfaces in the real projective space ${\mathbb{RP}}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces ${\mathbb{RP}}^k\subset {\mathbb{RP}}^n$ (starting with points). This confirms a conjecture of Burago and Zalgaller from 1988 and extends to higher dimensions previous result of M. Ritoré and A. Ros on ${\mathbb{RP}}^3$. We also derive an Willmore type inequality for antipodal invariant hypersurfaces in ${\mathbb{S}}^n$.

arXiv comment20 pages, 1 figure. Minor revisions. To appear in Journal of Differential Geometry