The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

Willmore Surfaces in S n

A surface x> : M S ⁿ is called a Willmore surface if it is a critical surface of the Willmore functional. It is well known that any minimal surface is a Willmore surface and that many nonminimal Willmore surfaces exists. In this paper, we establish an integral inequality for compact Willmore surfaces in S ⁿ and obtain a new characterization of the Veronese surface in S ⁴ as a Willmore surface. Our result reduces to a well-known result in the case of minimal surfaces.     

Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature

Let be a hypersurface in the (m+1)-dimensional unit sphere S^m+1 without umbilics. Four basic invariants of x under the Möbius transformation group in S^m+1 are a Riemannian metric g called Möbius metric, a 1-form called Möbius form, a symmetric (0,2) tensor A called Blaschke tensor and symmetric (0,2) tensor B called Möbius second fundamental form. In this paper, we prove the following classification theorem: let be a hypersurface, which satisfies (i) 0, (ii) A+g+B0 for some functions and , then and must be constant, and x is Möbius equivalent to either (i) a hypersurface with constant mean curvature and scalar curvature in S^m+1; or (ii) the pre-image of a stereographic projection of a hypersurface with constant mean curvature and scalar curvature in the Euclidean space R^m+1; or (iii) the image of the standard conformal map of a hypersurface with constant mean curvature and scalar curvature in the (m+1)-dimensional hyperbolic space H^m+1. This result shows that one can use Möbius differential geometry to unify the three different classes of hypersurface with constant mean curvature and scalar curvature in S^m+1, R^m+1 and H^m+1.Partially supported the Alexander Humboldt Stiftung and Zhongdian grant of NSFC.Partially supported by RFDP, Qiushi Award, 973 Project and Jiechu grant of NSFC.Mathematics Subject Classification (2000):Primary 53A30; Secondary 53B25     

Lagrangian spheres in the 2-dimensional complex space forms

By constructing a holomorphic cubic form for Lagrangian surfaces with nonzero constant length mean curvature vector in a 2-dimensional complex space form (4c), we characterize the Lagrangian pesudosphere as the only branched Lagrangian immersion of a sphere in (4c) with nonzero constant length mean curvature vector. When c = 0, our result reduces to Castro-Urbano’s result in [1]. H. Li is partially supported by NSFC grant No. 10531090 H. Ma is partially supported by NSFC grant No. 10501028 and SRF for ROCS, SEM     

Stability of area-preserving variations in space forms

In this article, we deal with compact hypersurfaces without boundary immersed in space forms with . They are critical points for an area-preserving variational problem. We show that they are r-stable if and only if they are totally umbilical hypersurfaces.      

A class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold

Science China Mathematics - In this paper, we study the star-shaped hypersurfaces evolved by a class of inverse mean curvature type flows in the anti-de Sitter-Schwarzschild manifold. We give C0,...     

A Moebius characterization of Veronese surfaces in $S^n$

Let be an umbilic-free submanifold in with I and II as the first and second fundamental forms. An important Moebius invariant for in Moebius differential geometry is the so-called Moebius form , defined by , where is a local basis of the tangent bundle with dual basis , is a local basis of the normal bundle, is the mean curvature vector and . In this paper we prove that if is an umbilics-free immersion of 2-sphere with vanishing Moebius form , then there exists a Moebius transformation and a 2k-equator with such that is the Veronese surface. Received August 12, 1999 / Published online March 12, 2001     

A variational problem for submanifolds in a sphere

Let be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S ^{n + p} , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional , where is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S ^{n + p} . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.     

Classification of Möbius Isoparametric Hypersurfaces in ${\Bbb S}^5$

Let M ⁿ be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere , then M ⁿ is associated with a so-called M?bius metric g, a M?bius second fundamental form B and a M?bius form Φ which are invariants of M ⁿ under the M?bius transformation group of . A classical theorem of M?bius geometry states that M ⁿ (n ≥ 3) is in fact characterized by g and B up to M?bius equivalence. A M?bius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hypersurfaces are automatically M?bius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, we determine all M?bius isoparametric hypersurfaces in by proving the following classification theorem: If is a M?bius isoparametric hypersurface, then x is M?bius equivalent to either (i) a hypersurface having parallel M?bius second fundamental form in ; or (ii) the pre-image of the stereographic projection of the cone in over the Cartan isoparametric hypersurface in with three distinct principal curvatures; or (iii) the Euclidean isoparametric hypersurface with four principal curvatures in . The classification of hypersurfaces in with parallel M?bius second fundamental form has been accomplished in our previous paper [7]. The present result is a counterpart of the classification for Dupin hypersurfaces in up to Lie equivalence obtained by R. Niebergall, T. Cecil and G. R. Jensen. Partially supported by DAAD; TU Berlin; Jiechu grant of Henan, China and SRF for ROCS, SEM. Partially supported by the Zhongdian grant No. 10531090 of NSFC. Partially supported by RFDP, 973 Project and Jiechu grant of NSFC.