Colloquium


DATE2018-06-21 16:10-17:00

PLACE數學館3174教室

SPEAKER鄭日新 研究員(中央研究院數學研究所

TITLEComplex/CR duality (?)—Volume Renormalization and Invariant Geometric Equations

ABSTRACT Inspired by submanifold observables in AdS/CFT correspondence, we consider volume renormalization of surfaces or hypersurfaces with boundary curves or surfaces in possible complex/CR duality. Precisely I will talk about biholomorphically invariant curves and surfaces on the boundary of a strongly pseudoconvex domain in C^2. A distinguished class of such invariant curves satisfies a system of 2nd order ODEs, called chains in CR geometry. We interpret chains as geodesics of a Kropina metric in Finsler geometry. The associated energy functional of a curve on the boundary can be recovered as the log term coefficient in a weighted renormalized area expansion of a minimal surface that it bounds inside the domain. For surfaces on the boundary, we express two CR invariant surface area elements in terms of quantities in pseudohermitian geometry. We deduce the Euler-Lagrange equations of the associated energy functionals. In relation to the singular CR Yamabe problem, we show that one of the energy functionals appears as the coefficient (up to a constant multiple) of the log term in the associated volume renormalization. We ask how these “CR Willmore” surfaces are related to geometric quantities inside the domain.