Let x : M m ! M, m 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifold M with sectional curvature satisfying k 2 K M 0, for some constant k. Assume that the immersion has nite total curvature in the sense that the traceless second fundamental form has nite L m -norm. If K M 6 0, assume further that the rst eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has nite dimension. Moreover there exists a constant > 0, explicitly computed, such that if the total curvature is bounded from above by then there is no nontrivial L 2 -harmonic 1-forms on M.

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Let Ν n+1 be a Riemannian manifold with sectional curvatures uniformly bounded from below. When n = 3,4, we prove that there are no complete (strongly) stable H-hypersurfaces, without boundary, provided |H| is large enough. In particular, we prove that there are no complete strongly stable H-hypersurfaces in R n+1 without boundary, H ≠ 0.

/pdf/stable-constant-mean-curvature-hypersurfaces-103dpw8irx.pdf

Stable constant mean curvature hypersurfaces

Vanishing theorems for L2 harmonic 1-forms on complete submanifolds in a Riemannian manifold

Vanishing theorems for L2 harmonic forms on complete submanifolds in Euclidean space

Let M m , with m � 3, be an m-dimensional complete noncompact manifold isometrically immersed in a Hadamard manifold ¯ M. Assume that the mean curvature vector has finite L p -norm, for some 2 � pm. We prove that each end of M must either have finite volume or be non-parabolic.

/pdf/the-non-parabolicity-of-infinite-volume-ends-1kkwmzhfx7.pdf

The non-parabolicity of infinite volume ends

Let Mn (n ≥ 3) be an n-dimensional complete noncompact oriented submanifold in an (n+p)-dimensional Euclidean space Rn+p with finite total mean curvature, i.e, ∫M|H|n < ∞, where H is the mean curvature vector of M. Then we prove that each end of M must be non-parabolic. Denote by φ the traceless second fundamental form of M. We also prove that if ∫M|φ|n < C (n), where C (n) is an an explicit positive constant, then there are no nontrivial L2 harmonic 1-forms on M and the first de Rham's cohomology group with compact support of M is trivial. As corollaries, such a submanifold has only one end. This implies that such a minimal submanifold is plane.

https://projecteuclid.org/download/pdf_1/euclid.kmj/1257948888

L2 harmonic 1-forms on complete submanifolds in Euclidean space

Let Mn (n ≤ 7) be an n-dimensional complete immersed super stable minimal submanifold in an (n + p)-dimensional Euclidean space Rn+p with flat normal bundle. We prove that if the second fundamental form A of M satisfies ∫M |A|3 < ∞, then M is an affine n-dimensional plane.

/pdf/minimal-submanifolds-with-flat-normal-bundle-1yf16997dc.pdf

Hai-Ping Fu

Papers

L2 harmonic 1-forms on complete submanifolds in Euclidean space

Minimal submanifolds with flat normal bundle