Degree Theory of Immersed Hypersurfaces

Harold Rosenberg; Graham Smith

Degree Theory of Immersed Hypersurfaces

Differential Geometry 2016-10-11 v3

Authors: Harold Rosenberg , Graham Smith

Abstract

We develop a degree theory for compact immersed hypersurfaces of prescribed $K$ -curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function. We apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where $K$ is mean curvature; extrinsic curvature and special Lagrangian curvature, and we show that in all these cases, this number is equal to $-\chi(M)$ , where $\chi(M)$ is the Euler characteristic of $M$ .

Keywords

degree theory immersed hypersurfaces K-curvature elliptic curvature immersed hyperspheres Euler characteristic

Cite

@article{arxiv.1010.1879,
  title  = {Degree Theory of Immersed Hypersurfaces},
  author = {Harold Rosenberg and Graham Smith},
  journal= {arXiv preprint arXiv:1010.1879},
  year   = {2016}
}

Comments

Complete revision. We've worked hard to make the text clearer and we hope the reader will notice the difference. Shorter and more conceptual proofs. The applications, we hope, are much clearer. In addition, we develop a new concept of "weakly smooth manifolds". This provides a nice smooth manifold structure for the space of unparametrised immersions (details in the introduction)

Degree Theory of Immersed Hypersurfaces

Abstract

Keywords

Cite

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