Compact submanifolds in space forms

We use closed conformal vector fields in a constant sectional curvature Riemannian manifold ${\mathbb{M}}$ to study the geometry of its immersed submanifolds. In this situation we obtain a characterization of sphere among compact submanifolds with positive Ricci curvature immersed in ${\mathbb{M}}$ .     

Real hypersurfaces of a complex space form

In this paper, we show that an n-dimensional connected non-compact Ricci soliton isometrically immersed in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ , with potential vector field of the Ricci soliton is the characteristic vector field of the real hypersurface is an Einstein manifold. We classify connected Hopf hypersurfaces in the flat complex space form ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle )}$ and also obtain a characterization for the Hopf hypersurfaces in ${(C^{\frac{n+1}{2}},J,\left\langle ,\right\rangle ) }$ .     

Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ^∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.     

The Gauss�CKronecker curvature of minimal hypersurfaces in four-dimensional space forms

Let ${{\mathbb{Q}^4}(c)}$ be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss?CKronecker curvature of a complete minimal hypersurface in ${\mathbb{Q}^4(c), c\leq 0}$ , whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M ³ of a space form ${{\mathbb{Q}^4}(c)}$ with constant Gauss?CKronecker curvature K. For the case c ?? 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of ${\mathbb{Q}^4(c)}$ with K constant.     

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.     

On the structure of complete hypersurfaces in a Riemannian manifold of nonnegative curvature and L 2 harmonic forms

Let M ⁿ be a complete oriented noncompact hypersurface in a complete Riemannian manifold N ⁿ⁺¹ of nonnegative sectional curvature with ${2 \leq n \leq 5}$ . We prove that if M satisfies a stability condition, then there are no non-trivial L ² harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of ${\mathbb{R}}$ and a compact manifold with nonnegative sectional curvature. In case ${n \geq 5}$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.     

Weakly Convex Biharmonic Hypersurfaces in Nonpositive Curvature Space Forms are Minimal

A submanifold M ^m of a Euclidean space R ^m+p is said to have harmonic mean curvature vector field if ${\Delta \vec{H}=0}$ , where ${\vec{H}}$ is the mean curvature vector field of ${M\hookrightarrow R^{m+p}}$ and Δ is the rough Laplacian on M. There is a famous conjecture named after Bangyen Chen which states that submanifolds of Euclidean spaces with harmonic mean curvature vector fields are minimal. In this paper we prove that weakly convex hypersurfaces (i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic mean curvature vector fields in Euclidean spaces are minimal. Furthermore we prove that weakly convex biharmonic hypersurfaces in nonpositively curved space forms are minimal.     

Surface diffusion flow near spheres

We consider closed immersed hypersurfaces evolving by surface diffusion flow, and perform an analysis based on local and global integral estimates. First we show that a properly immersed stationary (ΔH ≡ 0) hypersurface in ${\mathbb{R}^3}$ or ${\mathbb{R}^4}$ with restricted growth of the curvature at infinity and small total tracefree curvature must be an embedded union of umbilic hypersurfaces. Then we prove for surfaces that if the L ² norm of the tracefree curvature is globally initially small it is monotonic nonincreasing along the flow. We also derive pointwise estimates for all derivatives of the curvature assuming that its L ² norm is locally small. Using these results we show that if a singularity develops the curvature must concentrate in a definite manner, and prove that a blowup under suitable conditions converges to a nonumbilic embedded stationary surface. We obtain our main result as a consequence: the surface diffusion flow of a surface initially close to a sphere in L ² is a family of embeddings, exists for all time, and exponentially converges to a round sphere.     

Variational formulas of higher order mean curvatures

In this paper,we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional M2p of a submanifold M n in a general Riemannian manifold N n+m for p = 0,1,...,[n 2 ].As an example,we prove that closed complex submanifolds in complex projective spaces are critical points of the functional M2p,called relatively 2p-minimal submanifolds,for all p.At last,we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms,as well as a special variational problem.     

Willmore hypersurfaces with constant Möbius curvature in R n+1

For an immersed hypersurface ${f : M^n \rightarrow R^{n+1}}$ without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces ${(n \geq 3)}$ has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in ${S^{3} \subset R^{4}}$ . Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension ${n \ge 4}$ to n = 3, showing that they are cones over the homogeneous torus ${S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}$ , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R ², S ², H ², respectively.     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).     

Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

We investigate the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and prove that if M ⁿ is simply connected and the k-th Ricci curvature of M ⁿ is bounded below by a quantity involving the mean curvature of M ⁿ and the curvature of the ambient manifold, then M ⁿ is diffeomorphic to the standard sphere ${\mathbb{S}^n}$ . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold M ⁿ is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.     

The classification of (m, \rho)-quasi-Einstein manifolds

If there exist a smooth function f on $(M^n, g)$ and three real constants $m,\rho ,\lambda $ ( $0<m\le \infty $ ) such that $$\begin{aligned} R_{ij}+f_{ij}-\frac{1}{m}f_if_j=(\rho R+\lambda ) g_{ij}, \end{aligned}$$ we call $(M^n,g)$ a $(m,\rho )$ -quasi-Einstein manifold. Here $R_{ij}$ is the Ricci curvature and R is the scalar curvature of the metric g, respectively. This is a special case of the so-called generalized quasi-Einstein manifold which was a natural generalization of gradient Ricci solitons associated with the Hamilton’s Ricci flow. In this paper, we first obtain some rigidity results for compact $(m,\rho )$ -quasi-Einstein manifolds. Then, we give some classifications under the assumption that the Bach tensor of $(M^n,g)$ is flat.     

On closed minimal hypersurfaces with constant scalar curvature in \mathbb{S}^7

We consider minimal closed hypersurfaces ${M \subset \mathbb{S}^7(1)}$ with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.     

Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Following Li and Yau (Acta Math 156:153?C201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional ${\mathcal{J}}$ associated to a smooth function ${\phi}$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

Some pinching theorems for minimal submanifolds in \mathbb{S}^m (1) \times \mathbb{R}

Let Mn be an n-dimensional compact minimal submanifolds in Sm(1)×R.We prove two pinching theorems by the Ricci curvature and the sectional curvature pinching conditions respectively.In fact,we characterize the Clifford tori and Veronese submanifolds by our pinching conditions respectively.     

Stability index jump for constant mean curvature hypersurfaces of spheres

It is known that the totally umbilical hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. That is, a compact hypersurface with constant mean curvature, cmc, in S ⁿ⁺¹, different from an Euclidean sphere, must have stability index greater than or equal to 1. In this paper we prove that the weak stability index of any non-totally umbilical compact hypersurface ${M \subset S^{{n+1}}}$ with cmc cannot take the values 1, 2, 3 . . . , n.     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

Closed geodesics on Finsler spheres

In this paper, we prove that for every Finsler n-sphere (S ⁿ ,?F) all of whose prime closed geodesics are non-degenerate with reversibility λ and flag curvature K satisfying ${\left(\frac{\lambda}{\lambda+1}\right)^2 < K \le 1,}$ there exist ${2[\frac{n+1}{2}]-1}$ prime closed geodesics; moreover, there exist ${2[\frac{n}{2}]-1}$ non-hyperbolic prime closed geodesics provided the number of prime closed geodesics is finite.