Rigidity and stability of Einstein metrics for quadratic curvature\n functionals

2011/05/23 by Matthew J. Gursky, Jeff Viaclovsky, Gursky, Matthew +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #Dermatological and Skeletal Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1105.4648

openalex publication_date 2011/05/23 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We investigate rigidity and stability properties of critical points of\nquadratic curvature functionals on the space of Riemannian metrics. We show it\nis possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion,\nto become elliptic. Fredholm theory may then be used to describe local\nproperties of the moduli space of critical metrics. We show a number of compact\nexamples are infinitesimally rigid, and consequently, are isolated critical\npoints in the space of unit-volume Riemannian metrics. We then give examples of\ncritical metrics which are strict local minimizers (up to diffeomorphism and\nscaling). A corollary is a local "reverse Bishop's inequality" for such\nmetrics. In particular, any metric g in a C^2,\α-neighborhood of the\nround metric (Sⁿ,g_S) satisfying Ric(g) \≤ Ric(g_S) has volume Vol(g)\n\≥ Vol(g_S), with equality holding if and only if g is isometric to g_S.\n

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