Past Events

Nonlinear Analysis

C^2 Interior estimate for convex solutions of prescribing scalar curvature equation

Pengfei Guan, McGill University

Location:  Hill 705
Date & time: Tuesday, 06 March 2018 at 1:40PM - 2:40PM

Abstract:  A classical result of Heinz states that, if a convex graph in $$R^3$$ with positive smooth Gauss curvature, then there is an interior $$C^2$$ estimate.  The problem is reduced to $$2$$-dimensional Monge-Ampere equation.  Such interior estimate is false for Monge-Ampere equation when dimension is larger than or equal to $$3$$, by Pogorelov's example.  The work of Urbas also indicates the failure of interior estimate for $$\sigma_k$$ curvature and Hessian equations when $$k>2$$.  A longstand problem is where interior estimate is true for $$k=2$$ in general dimensions. In the talk, we provide an affirmative answer for all convex solutions.  In case of isometrically immersed hypersurfaces in $$R^{n+1}$$  with positive scalar curvature, convexity assumption can be dropped. These estimates are consequences of an interior estimates for these equations obtained under a weakened condition.

This is a joint work with Guohuan Qiu.

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