536 research outputs found
Interior estimate for Hessian quotient equation in dimension three
In this paper, we establish an interior estimate for the Hessian
quotient equation in dimension
three. A crucial ingredient in our proof is a Jacobi inequality
Variation and rigidity of quasi-local mass
Inspired by the work of Chen-Zhang \cite{Chen-Zhang}, we derive an evolution
formula for the Wang-Yau quasi-local energy in reference to a static space,
introduced by Chen-Wang-Wang-Yau \cite{CWWY}. If the reference static space
represents a mass minimizing, static extension of the initial surface ,
we observe that the derivative of the Wang-Yau quasi-local energy is equal to
the derivative of the Bartnik quasi-local mass at .
Combining the evolution formula for the quasi-local energy with a localized
Penrose inequality proved in \cite{Lu-Miao}, we prove a rigidity theorem for
compact -manifolds with nonnegative scalar curvature, with boundary. This
rigidity theorem in turn gives a characterization of the equality case of the
localized Penrose inequality in -dimension.Comment: new notations added; references updated; section 4 revise
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