Rigidity and non-rigidity results for conformal immersions

Rigidity and Non-rigidity Results on the Sphere

It is a simple consequence of the maximum principle that a superharmonic function u on Rn(i. e. ∆u ≤ 0) which is 1 near infinity is identically 1 on Rn (throughout this paper, n ≥ 3). Geometrically this means that one can not conformally deform the Euclidean metric in a bounded region without decreasing the scalar curvature somewhere. In fact there is a much stronger result: one can not have an...

Geometric Rigidity of Conformal Matrices

We provide a geometric rigidity estimate à la Friesecke-James-Müller for conformal matrices. Namely, we replace SO(n) by a arbitrary compact subset of conformal matrices, bounded away from 0 and invariant under SO(n), and rigid motions by Möbius transformations.

On Isometric and Conformal Rigidity of Submanifolds

Let f, g : Mn → Rn+d be two immersions of an n-dimensional differentiable manifold into Euclidean space. That g is conformal (isometric) to f means that the metrics induced on Mn by f and g are conformal (isometric). We say that f is conformally (isometrically) rigid if given any other conformal (isometric) immersion g there exists a conformal (isometric) diffeomorphism Υ from an open subset of...

Rigidity and Inflexibility in Conformal Dynamics

Institute for Mathematical Physics Rigidity for Local Holomorphic Conformal Embeddings from B Rigidity for Local Holomorphic Conformal Embeddings from B

In this article, we study local holomorphic conformal embeddings from B into B1 × · · · ×BNm with respect to the normalized Bergman metrics. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic...