In this note, we investigate the well-known Yau rigidity theorem for minimal submanifolds in spheres. Using the parameter method of Yau and the DDVV inequality verified by Lu, Ge and Tang, we prove that if $M$ is an $n$-dimensional oriented compact minimal submanifold in the unit sphere $S^{n+p}(1)$, and if $K_{M}\geq\frac{sgn(p-1)p}{2(p+1)},$ then $M$ is either a totally geodesic sphere, the standard immersion of the product of two spheres, or the Veronese surface in $S^4(1)$. Here $sgn(\cdot)$ is the standard sign function. We also extend the rigidity theorem above to the case where $M$ is a compact submanifold with parallel mean curvature in a space form.
In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.
We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.
New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.
It is known that there exist Calabi-Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we describe the Calabi-Yau structures of the complexified symmetric spaces in terms of the Schwarz's theorem in detail. We consider the case where the Calabi-Yau structure arises from the Riemannian metric corresponding to the Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi-Yau structure, we give constructions of special Lagrangian submanifolds of any phase which are invariant under the actions of symmetric subgroups of the isometry gr…
We show that an $n$-dimensional surface whose entropy is close to that of an $n$-dimensional plane is close in Hausdorff distance to some $n$-dimensional plane at every scale. Moreover we show that self-expanders of low entropy converge in the Hausdorff sense to their asymptotic cones.
In this paper we establish conditions on the length of the second fundamental form of a complete minimal submanifold $M^n$ in the hyperbolic space $\mathbb{H}^{n+m}$ in order to show that $M^n$ is totally geodesic. We also obtain sharp upper bounds estimates for the first eigenvalue of the super stability operator in the case of $M$ is a surface in $\mathbb{H}^{4}$.
In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.
A contact stationary Legendrian submanifold (briefly, CSL submanifold) is a stationary point of the volume functional of Legendrian submanifolds in a Sasakian manifold. Much effort has been paid in the last two decades to construct examples of such manifolds, mainly by geometers using various geometric methods. But we have rare knowledge about their geometric properties till now. Recently, Y. Luo (\cite{ Luo2, Luo1}) proved that a closed CSL surface in $\mathbb{S}^5$ with the square length of its second fundamental form belonging to $[0,2]$ must be totally geodesic or be a flat minimal Legendr…
A compact approach to the positivity of Brown-York's mass and its relation with the Min-Oo conjecture, Yau's Problem \#100 and rigidity of hypersurfaces
Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2σ_n}{2n-σ_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $σ_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theor…
We show a method to construct a special Lagrangian submanifold L' from a given special Lagrangian submanifold L in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in the cotangent bundles of the spheres which are non-flat Calabi-Yau manifolds equipped with the Stenzel metrics.
Rigidity of submanifolds with parallel mean curvature in space froms
设$M$是具有$c+H^2>0$的单连通空间形式$F^{n+p}(c)$中具有平行中曲率的$n(\geq3)$维定向紧致子流形，其中$H$是$M$的中曲率.证明了若$M$的Ricci曲率满足$Ric_{M}\geq(n-2)(c+H^2),$，则$M$要么是全脐球面，要么是$S^{n+1}(\frac{1}{\sqrt{c+H^2}})$ with $n=2m$中的Clifford超曲面$S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)\times S^{m}\big(\frac{1}{\sqrt{2(c+H^2)}}\big)$，要么是$S^7(\frac{1}{\sqrt{c+H^2}})$中的$\mathbb{C}P^{2}(4/3(c+H^2))$.特别地，如果$Ric_{M}>(n-2)(c+H^2),$，则$M$是完全脐球。
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
Rigidity of Lagrangian submanifolds with conformal Maslov form and Legendrian capillary boundary
Beijing University of Civil Engineering and Architecture（北京建筑大学）；Chongqing University of Technology（重庆理工大学）；Tsinghua University（清华大学）；Xinyang Normal University（信阳师范大学）
In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove that, if $c+H^2>0$ and $\|\mathrm{Ric}_{-}^λ\|_{n/2}< ε(n,c, λ, H)$ for $λ$ satisfying $ \frac{n-2}{n-1} (c+H^2) < λ\le c+H^2$, then $M$ is the totally umbilical sphere $\mathbb{S}^n(\tfrac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel mean curvature of $M$, and $ε(n,c,λ, H)$ is a positive constant depending only on $n, c,λ$ and $H$. This extends some of the earl…
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for sub…
We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are …
In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by …
New general results of non-existence and rigidity of spacelike submanifolds immersed in a spacetime, whose mean curvature is a time-oriented causal vector field, are given. These results hold for a wide class of spacetimes which includes globally hyperbolic, stationary, conformally stationary and pp-wave spacetimes, among others. Moreover, applications to the Cauchy problem in General Relativity, are presented. Finally, in the case of hypersurfaces, we also obtain significant consequences in Geometrical Analysis, solving new Calabi-Bernstein and Dirichlet problems on a Riemannian manifold.
Rigidity results for non-Kähler Calabi-Yau geometries on threefolds
Département de Mathématiques UQAM and Institute of Mathematics and Informatics Bulgarian Academy of Sciences（魁北克大学蒙特利尔分校数学系和保加利亚科学院数学与信息学研究所）；Department of Mathematics, Aarhus University（奥胡斯大学数学系）；Rowland Hall University of California, Irvine（加州大学欧文分校罗兰大厅）
