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Xingxiao Li

Publications and source records attributed to Xingxiao Li.

At least 19 recordsLinked to original sources
arXiv2020

On the mean curvature flow of submanifolds in the standard Gaussian space $^†$

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ where $({\mathbb R}^{m+p},\ol g)$ is the standard Euclidean space and $x\in{\mathbb R}^{m+p}$ denotes the position vector. Note that, as a special Riemannian manifold, $({\mathbb R}^{m+p},e^{-|x|^2/m}\ol g)$ has an unbounded curvature. Up to a family of diffeomorphisms on $M^m$, the mean curvature flow we considered here turns out to be equivalent to a special variation of the ``{\em conformal mean curvature flow}\,'' which we have introduced previously. The main theorem of this paper indicates, geometrically, that any immersed compact submanifold in the standard Gaussian space, with the square norm of the position vector being not equal to $m$, will blow up at a finite time under the mean curvature flow, in the sense that either the position or the curvature blows up to infinity; Moreover, by this main theorem, the interval $[0,T)$ of time in which the flowing submanifolds keep regular has some certain optimal upper bound, and it can reach the bound if and only if the initial submanifold either shrinks to the origin or expands uniformly to infinity under the flow. Besides the main theorem, we also obtain some other interesting conclusions which not only play their key roles in proving the main theorem but also characterize in part the geometric behavior of the flow, being of independent significance.

math.DG↗
arXiv2018

Rigidity of Einstein metrics as critical points of some quadratic curvature functionals on complete manifolds

In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the trace-less Ricci curvature and the Sobolev constant, accordingly.

math.DG↗
arXiv2018

Equiaffine isoparametric functions and their regular level hypersurfaces

In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$, we first introduce the concept of equiaffine parallel hypersurfaces in $A^{n+1}$, obtaining some fundamental identities with the basic equiaffine geometric invariants, and then we define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces of constant affine mean curvature in $A^{n+1}$. Finally, we introduce the concept of equiaffine isoparametric functions on $A^{n+1}$, and prove that any equiaffine isoparametric hypersurface is exactly a regular level set of some equiaffine isoparametric function.

math.DG↗
arXiv2018

The blow-up of the conformal mean curvature flow

In this paper, we introduce and study the conformal mean curvature flow of submanifolds of higher codimension in the Euclidean space $\bbr^n$. This kind of flow is a special case of a general modified mean curvature flow which is of various origination. As the main result, we prove a blow-up theorem concluding that, under the conformal mean curvature flow in $\bbr^n$, the maximum of the square norm of the second fundamental form of any compact submanifold tends to infinity in finite time. Furthermore, by using the idea of Andrews and Baker for studying the mean curvature flow of submanifolds in the Euclidean space, we also derive some more evolution formulas and inequalities which we believe to be useful in our further study of conformal mean curvature flow. Presently, these computations together with our main theorem are applied to provide a direct proof of a convergence theorem concluding that the external conformal forced mean curvature flow of a compact submanifold in $\bbr^n$ with the same pinched condition as Andrews-Baker's will be convergent to a round point in finite time.

math.DG↗
arXiv2016

Variational characterizations of $ξ$-submanifolds in the Eulicdean space $\bbr^{m+p}$

$ξ$-submanifold in the Euclidean space $\bbr^{m+p}$ is a natural extension of the concept of self-shrinker to the mean curvature flow in $\bbr^{m+p}$. It is also a generalization of the $λ$-hypersurface defined by Q.-M. Cheng et al to arbitrary codimensions. In this paper, some characterizations for $ξ$-submanifolds are established. First, it is shown that a submanifold in $\bbr^{m+p}$ is a $ξ$-submanifold if and only if its modified mean curvature is parallel when viewed as a submanifold in the Gaussian space $(\bbr^{m+p},e^{-\fr{|x|^2}{m}}\lagl\cdot,\cdot\ragl)$; Then, two weighted volume functionals $V_ξ$ and $\bar V_ξ$ are introduced and it is proved that $ξ$-submanifolds can be characterized as the critical points of these two functionals; Also, the corresponding second variation formulas are computed and the ($W$-)stability properties for $ξ$-submanifolds are systematically studied. In particular, it is proved that $m$-planes are the only properly immersed, complete $W$-stable $ξ$-submanifolds with flat normal bundle under a technical condition. It would be interesting if this additional restriction could be removed.

math.DG↗
arXiv2015

On the immersed submanifolds in the unit sphere with parallel Blaschke tensor

As is known, the Blaschke tensor $A$ (a symmetric covariant $2$-tensor) is one of the fundamental Möbius invariants in the Möbius differential geometry of submanifolds in the unit sphere $\mathbb S^n$, and the eigenvalues of $A$ are referred to as the Blaschke eigenvalues. In this paper, we shall prove a classification theorem for immersed umbilic-free submanifolds in $\mathbb S^n$ with a parallel Blaschke tensor. For proving this classification, some new kinds of examples are first defined.

math.DG↗
arXiv2015

On the immersed submanifolds in the unit sphere with parallel Blaschke tensor II

As is known, the Blaschke tensor $A$ (a symmetric covariant $2$-tensor) is one of the fundamental Möbius invariants in the Möbius differential geometry of submanifolds in the unit sphere $\mathbb S^n$, and the eigenvalues of $A$ are referred to as the Blaschke eigenvalues. In this paper, we continue our job for the study on the submanifolds in $\bbs^n$ with parallel Blaschke tensors which we simply call {\em Blaschke parallel submanifolds} to find more examples and seek a complete classification finally. The main theorem of this paper is the classification of Blaschke parallel submanifolds in $\mathbb S^n$ with exactly three distinct Blaschke eigenvalues. Before proving this classification we define, as usual, a new class of examples.

math.DG↗
arXiv2015

On the regular space-like hypersurfaces in the de Sitter space ${\mathbb S}^{m+1}_{1}$ with parallel Blaschke tensors

In this paper, we use two conformal non-homogeneous coordinate systems, modeled on the de Sitter space ${\mathbb S}^{m+1}_1$, to cover the conformal space ${\mathbb Q}^{m+1}_1$, so that the conformal geometry of regular space-like hypersurfaces in $\mathbb{Q}^{m+1}_1$ is treated as that of hypersurfaces in ${\mathbb S}^{m+1}_1$. As a result, we give a complete classification of the regular space-like hypersurfaces (represented in the de Sitter space ${\mathbb S}^{m+1}_1$) with parallel Blaschke tensors.

math.DG↗
arXiv2015

A rigidity theorem of $ξ$-submanifolds in $\mathbb{C}^{2}$

In this paper, we first introduce the concept of $ξ$-submanifold which is a natural generalization of self-shrinkers for the mean curvature flow and also an extension of $λ$-hypersurfaces to the higher codimension. Then, as the main result, we prove a rigidity theorem for Lagrangian $ξ$-submanifold in the complex $2$-plane $\bbc^2$.

math.DG↗
arXiv2015

On the Lagrangian angle and the Kähler angle of immersed surfaces in the complex plane $\bbc^2$

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surfaces of constant Kähler angle, together with an application showing that the Maslov class of a compact self-shrinker surface with constant Kähler angle is generally non-vanishing. Secondly, we obtain two pinching results for the Kähler angle which imply rigidity theorems of self-shrinkers with Kähler angle under the condition that $\int_M |h|^2e^{-\frac{|x|^2}{2}}dV_M<\infty$, where $h$ and $x$ denote, respectively, the second fundamental form and the position vector of the surface.

math.DG↗
arXiv2014

A classification theorem of nondegenerate equiaffine symmetric hypersurfaces

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing classification theorem of the semi-simple real Jordan algebras with unity, a complete classification for the nondegenerate and locally equiaffine symmetric hypersurfaces with nonzero affine mean curvatures is established. As an direct application of the main theorems, we prove at the end of the paper a complete classification of nondegenerate hypersurfaces with parallel Fubini-Pick forms and nonzero affine mean curvatures.

math.DG↗
arXiv2014

On the equiaffine symmetric hyperspheres

We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the first author, we are able to provide an alternative proof for the classification theorem of the affine hypersurface with parallel Fubini-Pick forms, which has already been established by Z.J. Hu et al in a totally different way.

math.DG↗
arXiv2014

A new characterization of Calabi composition of hyperbolic affine hyperspheres

In this paper, we mainly prove a theorem with a corollary establishing two characterizations of the Calabi composition of hyperbolic hyperspheres, where the second characterization (i.e., the corollary) has been given via a dual correspondence theorem earlier but now we would like to use a very direct method. Note that Z.J. Hu, H.Z. Li and L. Vrancken also gave a characterization of the $2$-factor Calabi composition in a different manner.

math.DG↗