On reduced L2 cohomology of hypersurfaces in spheres with finite total curvature.

Peng Zhu

doi:10.1590/0001-3765201620150085

Abstract

In this paper, we prove that the dimension of the second space of reduced L2 cohomology of M is finite if is a complete noncompact hypersurface in a sphere 𝕊n+1and has finite total curvature (n≥3).

Highlights

For a complete manifold M n, the p-th space of reduced L2-cohomology is defined, for 0 ≤ p ≤ n in Carron (2007)
The reduced L2 cohomology is related with the L2 harmonic forms (Carron 2007)
Proposition 6. (Hoffman and Spruck 1974, Zhu and Fang 2014) Let M n be a complete noncompact oriented manifold isometrically immersed in a sphere Sn+1

Summary

INTRODUCTION

Cavalcante et al (2014) discussed a complete noncompact submanifold M n (n ≥ 3) isometrically immersed in a Hadamard manifold N n+p with sectional curvature satisfying −k2 ≤ KN ≤ 0 for some constant k and showed that if the total curvature is finite and the first eigenvalue of the Laplacian operator of M is bounded from below by a suitable constant, the dimension of the space of the L2 harmonic 1-forms on M is finite. (Carron 2007) Let (M, g) is a complete Riemannian manifold, the space of L2 harmonic p-forms Hp(L2(M )) is isomorphic to the p-th space of reduced L2 cohomology H2p(M ). (Hoffman and Spruck 1974, Zhu and Fang 2014) Let M n be a complete noncompact oriented manifold isometrically immersed in a sphere Sn+1. (n − 2) + (λ1 + · · · + λn)λi − λ2i − λiλj (aij)

Note that

PROOF OF MAIN RESULTS

Choose f p