Xin Zhou

Assistant Professor

Address: Department of Mathematics
               South Hall, Room 6501
               University of California Santa Barbara
               Santa Barbara, California 93106 USA
E-mail:   zhou "at" math.ucsb.edu

I am on leave and visiting the Institute for Advanced Study for the academic year 2018-2019.

I am a member of the Geometry group. Before moving to UCSB, I was a CLE Moore instructor in the Math department at MIT. I received my PhD from Stanford University in 2013 with Richard M. Schoen as my advisor. Here is my Curriculum Vitae. My research is supported by NSF Grant DMS-1811293 and UC Regent's Junior Faculty Award.

Research Interest:

Preprints:

  1. Existence of hypersurfaces with prescribed mean curvature I - Generic min-max (with Jonathan Zhu), arXiv:1808.03527.
  2. Min-max minimal disks with free boundary in Riemannian manifolds (with Longzhi Lin and Ao Sun), arXiv:1806.04664.
  3. Compactness and generic finiteness for free boundary minimal hypersurfaces (with Qiang Guang), arXiv:1803.01509.
  4. Free boundary minimal hypersurfaces with least area (with Qiang Guang and Zhichao Wang), arXiv:1801.07036.
  5. Min-max theory for CMC hypersurfaces (with Jonathan Zhu), arXiv:1707.08012.
  6. A maximum principle for free boundary minimal varieties of arbitrary codimension (with Martin Li), arXiv:1708.05001.
  7. Min-max theory for free boundary minimal hypersurfaces I - regularity theory (with Martin Li), arXiv:1611.02612.

Publications:

  1. Curvature estimates for stable free boundary minimal hypersurfaces (with Qiang Guang and Martin Li), accepted by J. Reine Angew. Math (Crelle's Journal), arXiv:1611.02605.
  2. Sweeping out 3-manifold of positive Ricci curvature by short 1-cycles via estimates of min-max surfaces (with Y. Liokumovich), Int. Math. Res. Not. IMRN 2018, no. 4, 1129-1152 .
  3. Entropy of closed surfaces and min-max theory (with D. Ketover), J. Differential Geom. 110 (2018), no. 1, 31-71.
  4. Existence of minimal surfaces of arbitrarily large Morse index (with Haozhao. Li), Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 64, 12 pp.
  5. Min-max hypersurface in manifold of positive Ricci curvature, J. Differential Geom. 105 (2017), no. 2, 291-343.
  6. On the free boundary min-max geodesics, Int. Math. Res. Not. IMRN 2016, no. 5, 1447-1466.
  7. Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\leq n\leq 6$, J. Differential Geom. 100 (2015), no. 1, 129-160.
  8. Mass angular momentum inequality for axisymmetric vacuum data with small trace, Comm. Anal. Geom. 22 (2014), no. 3, 519-571.
  9. Convexity of reduced energy and mass angular momentum inequalities (with R. Schoen), Ann. Henri Poincaré 14 (2013), no. 7, 1747-1773.
  10. On the existence of min-max minimal surfaces of genus g≥ 2, Commun. Contemp. Math. 19 (2017), no. 4, 1750041, 36 pp.
  11. On the existence of min-max minimal torus, J. Geom. Anal. 20 (2010), no. 4, 1026-1055.

Research reports and surveys:

  1. Min-max theory for constant mean curvature (CMC) hypersurfaces (with Jonathan Zhu), Partial Differential Equations, Oberwolfach Report No. 35/2017.
  2. On minimal surfaces with free boundary (with Martin Li), special issues of ICCM Notices, to appear.
  3. Recent progress on compactness of minimal surfaces with free boundary (with Qiang Guang), Surveys in Geometric Analysis 2017, 63-78, Science Press Beijing, Beijing, 2018. ISBN: 9787030573223.

Lecture notes:

Slides and videos:

Past Conferences:

Past Teaching: