Curvature estimates for minimal hypersurfaces in singular spaces

We here prove pointwise curvature estimates for minimal hypersurfaces in singular spaces, using the integral curvature estimate and a generalized Simons inequality which were established recently. A further basic ingredient is a new Sobolev type inequality for stationary hypersurfaces.Oblatum 19-IX-1994This research was supported by the Deutsche Forschungsgemeinschaft through a Heisenberg award. Part of the work described here was carried out during visits to Dipartimento Matematica Applicata, University of Firenze and Department of Mathematics, Stanford University. The author would like to express his gratitude to both institutions for their kind hospitality and support. Also it is a pleasure to thank the referee for his useful comments concerning the formulation of the main theorem.     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

Stably embedded minimal hypersurfaces

We use Schoen’s curvature estimates to prove that the subfocal tubular neighborhood of a nonplanar minimal hypersurface with bounded second fundamental form, stably embedded in whose radius decays sufficiently slowly cannot be embedded. In particular such hypersurfaces admit no embedded tubular neighborhoods of constant radius, whatever small the radius. However, assuming a further hypothesis on the embedding, we prove that such hypersurfaces admit an embedded tube whose radius decays sufficiently fast.      

Integral curvature estimates for F-stable hypersurfaces

We consider immersed hypersurfaces in euclidean 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 , 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 .Received: 14 July 2003, Accepted: 13 September 2004, Published online: 10 December 2004Mathematics Subject Classification: 53C42, 49Q10, 35J60     

Compactness analysis for free boundary minimal hypersurfaces

We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

New minimal hypersurfaces in and

We find a class of minimal hypersurfaces as the zero level set of Pfaffians, resp. determinants of real dimensional antisymmetric matrices. While and are congruent to the quadratic cone resp. Hsiang's cubic invariant in , (special harmonic ‐invariant cones of degree ⩾4) seem to be new.     

Positive scalar curvature and minimal hypersurfaces

We show that the minimal hypersurface method of Schoen and Yau can be used for the ``quantitative' study of positive scalar curvature. More precisely, we show that if a manifold admits a metric with or , where is the scalar curvature of , any 2-tensor on and the Weyl tensor of , then any closed orientable stable minimal (totally geodesic in the second case) hypersurface also admits a metric with the corresponding positivity of scalar curvature. A corollary pertaining to the topology of such hypersurfaces is proved in a special situation.