Research & Publications

Research interests: Geometric analysis and Partial differential equations


 

Preprints

[10] Mean curvature flow in Fuchsian manifolds (joint with Zheng Huang and Zhou Zhang), LANL(arXiv.org) link.

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.

[9] Mean curvature flow of star-shaped hypersurfaces, LANL(arXiv.org) link.

Abstract:  In the last 15 years,  the series of works of White and Huisken-Sinestrari  yield that the blowup limits at singularities are convex for the mean curvature flow of mean convex hypersurfaces. In 1998  Smoczyk showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in R^3. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in R^{n+1} in arbitrary dimension n bigger than 1. In fact, this holds for a much more general class of initial hypersurfaces. In particular, this implies that the mean curvature flow of star-shaped hypersurfaces is generic in the sense of Colding-Minicozzi. 

Publications

[8] Blow-up of the mean curvature at the first singular time of the mean curvature flow (joint with Natasa Sesum); LANL(arXiv.org) link, Calculus of Variations and PDEs, June 2016, 55:65.

Abstract: It is conjectured that the mean curvature blows up at the first singular time of the mean curvature flow in Euclidean space, at least in dimensions less or equal to 7. We show that the mean curvature blows up at the singularities of the mean curvature flow starting from an immersed closed hypersurface with small L^2-norm of the traceless second fundamental form (observe that the initial hypersurface is not necessarily convex). As a consequence of the proof of this result we also obtain the dynamic stability of a sphere along the mean curvature flow with respect to the L^2-norm.

[7] 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.

Abstract: We show that the surface area preserving mean curvature flow in Euclidean space exists for all time and converges exponentially to a round sphere, if initially the L^2-norm of the traceless second fundamental form is small (but the initial hypersurface is not necessarily convex).

[6] Uniformity of harmonic map heat flow at infinite time (Analysis & PDE, Vol. 6 (2013), No. 8, 1899–1921; A&PDE Link); LANL(arXiv.org) link.

Abstract: We show an energy convexity along the harmonic map heat flow with small initial energy and fixed boundary data on the unit 2-disk. In particular, this gives an affirmative answer to an open question asking whether such harmonic map heat flow converges uniformly in time and strongly in the W^{1,2}-topology to the unique limiting harmonic map as time goes to infinity.

[5] 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.

Abstract: We show that the energy density of critical points of a class of conformally invariant variational problems with small energy on the unit 2-disk B_1 lies in the local Hardy space h^1(B_1). As a corollary we obtain a new proof of the energy convexity and uniqueness result for weakly harmonic maps with small energy on B_1.

[4] Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space (joint with Ling Xiao, Comm. Anal. Geom., Volume 20, Number 5, 1061-1096, 2012; CAG Link); LANL(arXiv.org) link.

Abstract: We define a new version of modified mean curvature flow (MMCF) in hyperbolic space H^{n+1}, which interestingly turns out to be the natural negative L^2-gradient flow of the energy functional defined by De Silva and Spruck. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, which was first shown by Guan and Spruck.

[3] Closed geodesics in Alexandrov spaces of curvature bounded from above (J. Geom. Anal., Volume 21, Issue 2 (2011), 429-454; JGA Link); LANL(arXiv.org) link.

Abstract: We show a local energy convexity of W^{1,2} maps into CAT(K) spaces. This energy convexity allows us to extend Colding and Minicozzi's width-sweepout construction to produce closed geodesics in any closed Alexandrov space of curvature bounded from above, which also provides a generalized version of the Birkhoff-Lyusternik theorem on the existence of non-trivial closed geodesics in the Alexandrov setting.

[2] Existence of good sweepouts on closed manifolds (joint with Lu Wang, Proc. Amer. Math. Soc. 138 (2010), No. 11, 4081-4088; PROC AMS Link); LANL(arXiv.org) link.

Abstract: We use the harmonic map heat flow to tighten the sweepout of closed curves on a closed Riemannian manifold, and we show that the tightened sweepout has the following good property: each curve in the tightened sweepout, whose energy is close to the maximal energy of curves in the sweepout, is itself close to a closed geodesic.

[1] 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.