Integral curvature estimates for F-stable hypersurfaces

Abstract

We consider immersed hypersurfaces in euclidean R n+1 which are stable with respect to an elliptic parametric functional with integrand F = F (N ) depending on normal directions only. We prove an integral curvature estimate provided that F is sufficiently close to the area integrand, extending the classical estimate of Schoen, Simon and Yau [19] for stable minimal hypersurfaces in R n+1 , as well as the pointwise estimate of Simon [22] for F -minimizing hypersurfaces. As a crucial point of our analysis we derive a generalized Simons inequality for the laplacian of the length of a weighted second fundamental form with respect to an abstract metric associated with F . As an application, we obtain a new Bernstein result for complete F -stable hypersurfaces of dimension n ≤ 5.

Pointwise curvature estimates for F-stable hypersurfaces

“…In this section we recall some preliminary results from [25] concerning the geometry of parametric functionals. Let X : M n → R n+1 be a smooth immersion of an n-dimensional oriented manifold without boundary into euclidean R n+1 .…”

“…In a previous paper [25] we have shown that an integral curvature estimate for F -stable hypersurfaces can be proved, whenever F is sufficiently close to the area functional. To be precise, define the norm…”

“…The paper is organized as follows. In Section 2 we recall some preliminary results taken from [25], including a generalized Simons inequality and an equivalent form of the integral curvature estimate. In Section 3 we then use these results to establish an L p -estimate for the curvature.…”

On surfaces of prescribed weighted mean curvature

“…We denote by N : Ω → S n its normal vector. In recent papers such as [4,14,15] or [16] critical points of the specific, anisotropic parametric functional…”

A Bernstein result for entire F-minimal graphs

“…If B 2R (x 0 ) ⊂ , then µ(M ∩ B ρ (X 0 )) ≤ C(n, F)ρ n (4.2) for all ρ ∈ [0, R], where X 0 := (x 0 , u(x 0 )).Proof of Theorem 4.1:According to the definition of δ (n, γ) there exists a number p > n 1−γ , p > 4n, such that F − A C 3 < δ(n, p). Put M := graph(u), X 0 := (0, u(0)) and choose ϕ to be a cut-off function satisfyingϕ = 1 in M ∩ B R (X 0 ), supp(ϕ) ⊂ M ∩ B 2R (X 0 ), 0 ≤ ϕ ≤ 1 and ∇ ϕ F ≤ C(F ) R .Applying ϕ in the integral curvature estimate (3.1) we obtain in view of (4.2) and the fact that dµ F ≤ C(F)dµ, see[25, Proposition 2.3],M∩B R (X 0 ) |S F | p F v p dµ F ≤ C(n, F, p)R n− p sup M∩B 2R (X 0 ) v p .…”