Graham-Witten's conformal invariant for closed four dimensional submanifolds

2017/03/24 by Yongbing Zhang, Zhang, Yongbing · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1703.08611

arxiv created 2022/11/05 · arxiv updated 2022/11/08

Abstract

It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold Σ is contained in the volume expansion of the minimal surface which is asymptotic to Σ when the minimal surface approaches the conformaly infinity. In the paper we give the explicit expression of Graham-Witten's conformal invariant for closed four dimensional submanifolds and find critical points of the conformal invariant in the case of Euclidean ambient spaces.

Cited by

• Construction of harmonic coordinates for weak immersions
• Structural Equations for Critical Points of Conformally Invariant Curvature Energies in 4d
• Energy reduction for Fourth order Willmore energy

Related

• Volume and Area Renormalizations for Conformally Compact Einstein Metrics
• Conformal Geometry on Four Manifolds
• A New Conformal Invariant for four-Dimensional Hypersurfaces
• On the topology of conformally compact Einstein 4-manifolds
• An invariant related to the existence of conformally compact Einstein fillings