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Min‐Oo Conjecture for Fully Nonlinear Conformally Invariant Equations
Author(s) -
Barbosa Ezequiel,
Cavalcante Marcos Petrúcio,
Espinar José María
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21866
Subject(s) - geodesic , mathematics , mathematical analysis , rigidity (electromagnetism) , ball (mathematics) , nonlinear system , pure mathematics , bounded function , physics , quantum mechanics
In this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n ‐sphere S n under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with. This proves rigidity for compact, connected, locally conformally flat manifolds ( M , g ) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D ( r ) , where D ( r ) denotes a geodesic ball of radius r ∈ (0, π /2] in S n , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, ( M , g ) must be isometric to the closed geodesic ballD r ¯ . As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three‐space ℍ 3 . In fact, we extend it to obtain rigidity for supersolutions to certain Monge‐Ampère equations. © 2019 Wiley Periodicals, Inc.