53 citations · 204 across the 31 of their papers we have counts for
15 papers · 1 filter
Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces
Xiaobo Liu, Wanxu Yang
Mean curvature flows of isoparametric submanifolds in Euclidean spaces and spheres have been studied by Liu and Terng. In particular, it was proved that such flows always have anci…
Polar foliations on symmetric spaces and the mean curvature flow
Xiaobo Liu, Marco Radeschi
In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by He…
Ancient solutions to mean curvature flow for isoparametric submanifolds
Xiaobo Liu, Chuu-Lian Terng
Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are anci…
Genus-2 G-function for orbifolds
Xiaobo Liu, Xin Wang
In this paper we prove that for Gromov-Witten theory of orbifolds of ADE type the genus-2 G-function introduced by B. Dubrovin, S. Liu, and Y. Zhang vanishes. Together with o…
Conditions for the vanishing of the genus-2 G-function
Xiaobo Liu, Xin Wang
In this paper we give some sufficient conditions for the vanishing of the genus-2 G-function, which was introduced by B. Dubrovin, S. Liu and Y. Zhang in [DLZ]. As a corollary we p…
Genus-1 Virasoro conjecture along quantum volume direction
Xiaobo Liu
In this paper, we show that the derivative of the genus-1 Virasoro conjecture for Gromov-Witten invariants along the direction of quantum volume element holds for all smooth projec…
Topological Recursion Relations on
Takashi Kimura, Xiaobo Liu
In this paper, we give some new genus-3 universal equations for Gromov-Witten invariants of compact symplectic manifolds. These equations were obtained by studying new relations in…
On Certain Vanishing Identities For Gromov-Witten Invariants
Xiaobo Liu
In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range…
The mean curvature flow for isoparametric submanifolds
Xiaobo Liu, Chuu-Lian Terng
A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow…
A Genus-3 Topological Recursion Relation
Takashi Kimura, Xiaobo Liu
In this paper, we give a new genus-3 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds. This formula also applies to intersection numbers…
Genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology
Xiaobo Liu
In this paper, we give a simple formula for the generating function of genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology, and use this formula to pr…
Idempotents on the big phase space
Xiaobo Liu
In this paper we study vector fields on the big phase space of Gromov-Witten theory which are idempotents of the quantum product. Such vector fields can be used to simplify univers…
Relations Among Universal Equations For Gromov-Witten Invariants
Xiaobo Liu
In this paper, we study relations among known universal equations for Gromov-Witten invariants at genus 1 and 2.
Isoparametric submanifolds and a Chevalley-type restriction theorem
Ernst Heintze, Xiaobo Liu, Carlos Olmos
We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Riemannian submersions…
Homogeneity of infinite dimensional isoparametric submanifolds
Ernst Heintze, Xiaobo Liu
A subset S of a Riemannian manifold N is called extrinsically homogeneous if S is an orbit of a subgroup of the isometry group of N. Thorbergsson proved the remarkable result that…