Vanishing theorems for $$L^{2}$$ L 2 harmonic forms on complete Riemannian manifolds

Abstract

This paper contains some vanishing theorems for L 2 harmonic forms on complete Riemannian manifolds with a weighted Poincaré inequality and a certain lower bound of the curvature. The results are in the spirit of Li-Wang and Lam, but without assumptions of sign and growth rate of the weight function, so they can be applied to complete stable hypersurfaces.

$L^p$ harmonic $1$-form on submanifold with weighted Poincaré inequality

“…Thereafter, using Bochner's vanishing technique, Miyaoka in [16] showed that a complete orientable noncompact stable minimal hypersurface in a Riemannnian manifold with nonnegative sectional curvature has no nontrivial L 2 harmonic 1-forms. Later, this result was extended to more general ambient spaces, see [12], [15], [24]. Seo in [19] proved the vanishing theorem holds for a complete stable minimal hypersurface in H n+1 with the first eigenvalue of the Laplacian satisfying λ 1 > (2n−1)(n−1).…”

“…ln/(n−l) dV (n−l)/n S(n, l) M (|∇h| l + (h|H|) l ) dV for all nonnegative C 1 -functions h : M n → R with compact support, where S(n, l) 1/l = c(n)2l(n − 1)/(n − l) and c(n) is a positive constant, depending only on n.The last but most important lemma was proved by Vieira in[24].…”

“…[24]). Let M be a complete manifold satisfying a weighted Poincaré inequality with a weight function ̺.…”

$L^p$ Harmonic $k$-Forms on Complete Noncompact Hypersurfaces in $\mathbb{S}^{n+1}$ with Finite Total Curvature

“…Since there are affluent finiteness results on reduced L 2 cohomology and L 2 harmonic forms both on manifolds and submanifolds, such as [2,5,6,8,12,13,16,17,19,22], and references therein. We want to investigate which kind of finiteness results can be generalized to L p case, and here we will concentrate on hypersurfaces in S n+1 .…”

Gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality

“…For further examples with applications to Riemannian geometry and hypersurface theory see [11], [24], [26], [37].…”