Research interests: Geometric analysis and Partial differential equations, more precisely, geometric flows including mean curvature flow and harmonic map heat flow, harmonic maps, minimal surfaces, surfaces of constant mean curvature, and min-max theory.
Abstract: In this paper, we establish a min-max theory for constructing minimal disks with free boundary in any closed Riemannian manifold. The main result is an effective version of the partial Morse theory for minimal disks with free boundary established by Fraser. Our theory also includes as a special case the min-max theory for Plateau problem of minimal disks, which can be used to generalize the famous work by Morse-Thompkins and Shiffman on minimal surfaces in R^n to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfaces using harmonic replacement introduced by Colding and Minicozzi to the free boundary setting. As a key ingredient to this construction, we show an energy convexity for weakly harmonic maps with mixed Dirichlet and free boundaries from the half unit 2-disk in R^2 into any closed Riemannian manifold, which in particular yields the uniqueness of such weakly harmonic maps. This is a free boundary analogue of the energy convexity and uniqueness for weakly harmonic maps with Dirichlet boundary on the unit 2-disk proved by Colding and Minicozzi.
Abstract: Every harmonic map is an intrinsic bi-harmonic map as an absolute minimizer of the intrinsic bi-energy functional, therefore intrinsic bi-harmonic map and its heat flow are more geometrically natural to study, but they are also considerably more difficult analytically than the extrinsic counterparts due to the lack of coercivity for the intrinsic bi-energy. In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball into the sphere S^n. This is a higher-order analogue of the energy convexity and uniqueness for weakly harmonic maps on unit 2-disk in R^2 proved by Colding and Minicozzi in 2008 (see also Lamm and the second author's work in 2012). In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. As an application, we also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into S^n, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm in 2005. Moreover, the energy convexity along the flow yields the uniform convergence of the flow which is not known before. One of the key ingredients in our proofs is a refined version of the ϵ-regularity of the first author and Riviere.
 Modified mean curvature flow of entire locally Lipschitz radial graphs in hyperbolic space (joint with Patrick Allmann and Jingyong Zhu), arXiv:1707.07087
Abstract: In a previous joint work of Xiao and the second author, the modified mean curvature flow (MMCF) in hyperbolic space was first introduced and the flow starting from an entire Lipschitz continuous radial graph with uniform local ball condition on the asymptotic boundary was shown to exist for all time and converge to a complete hypersurface of constant mean curvature with prescribed asymptotic boundary at infinity. In this paper, we remove the uniform local ball condition on the asymptotic boundary of the initial hypersurface, and prove that the MMCF starting from an entire locally Lipschitz continuous radial graph exists and stays radially graphic for all time.
Abstract: Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface Σ, such that the mean curvature flow exists for all time and converges to Σ. This is among the first examples of converging mean curvature flows of compact hypersurfaces in Riemannian manifolds. We also provide some useful calculations for the general warped product setting.
Published Journal Articles
 Mean curvature flow of star-shaped hypersurfaces, arXiv:1508.01225; to appear in Comm. Anal. Geom.
 Blow-up of the mean curvature at the first singular time of the mean curvature flow (joint with Natasa Sesum, Calculus of Variations and PDEs, June 2016, 55:65; Calc. Var. Link); LANL(arXiv.org) link,
 Stability of the surface area preserving mean curvature flow in Euclidean space (joint with Zheng Huang, Journal of Geometry, December 2015, Volume 106, Issue 3, 483-501; J. of Geom. Link); LANL(arXiv.org) link.
 Estimates for the energy density of critical points of a class of conformally invariant variational problems (joint with Tobias Lamm, Adv. Calc. Var., Volume 6, Issue 4 (Jul 2012), 391--413; ACV Link); LANL(arXiv.org) link.
 On the existence of closed geodesics and uniqueness of weakly harmonic maps, Thesis (Ph.D.), The Johns Hopkins University (2011), 76 pp. ISBN: 978-1124-75827-5.