University of Birmingham > Talks@bham > Analysis Seminar > Conformal Immersions into Riemannian Manifolds

Conformal Immersions into Riemannian Manifolds

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If you have a question about this talk, please contact Yuzhao Wang.

We develop the geometry and analysis of conformal immersions of surfaces into an a Riemannian manifold. We show an optimal Wente estimate for the Liouville Equation and as a consequence prove a singularity removal theorem for isolated singularities of conformal surfaces. This is used to extend a theorem of B. White to the setting of immersions of Riemannian manifolds. Next we prove a formula for the the Gauss Bonnet theorem and the conformal Willmore energy under conformal deformations of the ambient metric that generalises the conformal invariance of the Euclidean Willmore energy. A local compactness theorem is proved with optimal constants if n ≥ 4 and an energy identity is proved for the Gauss-Bonnet energy in the presence of a single bubble. This is used to obtain the optimal constant for n = 3. Finally we prove a geometric rigidity theorem for the generalised total curvature integral, showing that if we attain the optimal constant in the local compactness theorem but lose weak compactness then this is due to the presence of a complete compact minimal surface in Rn.

This talk is part of the Analysis Seminar series.

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