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Eigenvalue Estimates and Geometric Rigidity of Hypersurfaces

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GSTW05 - Emerging Horizons in Geometric Spectral Theory: an ECRs workshop

In this talk, I will begin with a brief review of some foundational geometric inequalities for hypersurfaces in Euclidean spaces focusing on those where equality charaacterizes the standared geodesic spheres. A prime example is Reilly’s celebraated inequality which provides a sharp upper bound for the first non trivial eigenvalue of the Laplace Beltrami-operator on compact, embedded hypersurfaces. Interestingly, these inequalities can often be traced back to a fundamental estimate involving the $L2$ norm of the position vector. I will then delve into a novel stability refinement of this inequality: we establish that if a hypersurface nearly attains this lower bound on the $L2$-norm, then it must be geometrically close, in a suitable sense, to a round sphere.

This talk is part of the Isaac Newton Institute Seminar Series series.

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