Preprint A705/2011
Degree Theory for Immersed Hypersurfaces

Graham Smith | Rosenberg, Harold

**Keywords: **
Degree Theory | Immersions | Convexity | Prescribed Curvature | Non-Linear Elliptic PDEs.

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 $M$.