On complete noncompact submanifolds with constant mean curvature and finite total curvature in Euclidean spaces

Abstract

In this paper, we consider a complete noncompact n-submanifold M with parallel mean curvature vector h in an Euclidean space. If M has finite total curvature, we prove that M must be minimal, so that M is an affine n-plane if it is strongly stable. This is a generalization of the result on strongly stable complete hypersurfaces with constant mean curvature in R n+1 .

REMARKS ON Lp-VANISHING RESULTS IN GEOMETRIC ANALYSIS

“…The upper bound m = 7 is due to the famous non-planar (and stable) minimal graphs by E. Bombieri, E. de Giorgi and E. Giusti. We also record that, in higher dimensions, condition (i) can be replaced by the next more general requirement, [35], [25],…”

Remarks on $L^{p}$-vanishing results in geometric analysis

“…The upper bound m = 7 is due to the famous non-planar (and stable) minimal graphs by E. Bombieri, E. de Giorgi and E. Giusti. We also record that, in higher dimensions, condition (i) can be replaced by the next more general requirement, [35], [25], (i)' |II| 2 ∈ L γ (M ), for some γ ≥ m/2. As we shall see in the next section, the approaches proposed by the above mentioned authors are very different from each others.…”

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

“…when n = 4, 2 < p < 3+ In fact, if (3.23) is true, we can apply Theorem 1.1 in do Carmo, Cheung, and Santos [11] or Corollary 4.3 in Xu and Deng [19] to show that M n is compact. Finally, a theorem in Barabosa, do Carmo, and Eschenburg [3] shows that the hypersurface is indeed a geodesic sphere.…”

“…And more recently, Xu and Deng [19] generalized it to higher codimensions. Now we shall generalize this result to complete stable minimal hypersurfaces in R n+1 with some finite L p norm curvature:…”