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

Current Teaching: Math 4B: Differential Equations.

We are organizing the UCSB Differential Geometry Seminar.

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.

Research Interest:

Awards and grants:

Preprints:

  1. Min-max theory for free boundary minimal hypersurfaces II -- General Morse index bounds and applications (with Q. Guang, M. Li, and Z. Wang), arXiv:1907.12064.
  2. On the Multiplicity One Conjecture in min-max theory, arXiv:1901.01173.
  3. Compactness and generic finiteness for free boundary minimal hypersurfaces (I) (with Q. Guang and Z. Wang), arXiv:1803.01509.
  4. Free boundary minimal hypersurfaces with least area (with Q. Guang and Z. Wang), arXiv:1801.07036.

Publications:

  1. Min-max theory for networks of constant geodesic curvature (with J. Zhu), accepted by Advances in Math., arXiv:1811.04070.
  2. Existence of hypersurfaces with prescribed mean curvature I - Generic min-max (with J. Zhu), accepted by Cambridge Journal of Mathematics, arXiv:1808.03527.
  3. Min-max minimal disks with free boundary in Riemannian manifolds (with L. Lin and A. Sun), accepted by Geometry & Topology, arXiv:1806.04664.
  4. Min-max theory for constant mean curvature hypersurfaces (with J. Zhu), Invent. math. (2019) 218:441-490.
  5. Min-max theory for free boundary minimal hypersurfaces I - regularity theory (with M. Li), accepted by J. Differential Geometry, arXiv:1611.02612.
  6. A maximum principle for free boundary minimal varieties of arbitrary codimension (with M. Li), accepted by Comm. Anal. Geom., arXiv:1708.05001.
  7. Curvature estimates for stable free boundary minimal hypersurfaces (with Q. Guang and M. Li), accepted by J. Reine Angew. Math. (Crelle's Journal), DOI: https://doi.org/10.1515/crelle-2018-0008, arXiv:1611.02605.
  8. 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 .
  9. Entropy of closed surfaces and min-max theory (with D. Ketover), J. Differential Geom. 110 (2018), no. 1, 31-71.
  10. Existence of minimal surfaces of arbitrarily large Morse index (with H. Li), Calc. Var. Partial Differential Equations 55 (2016), no. 3, Art. 64, 12 pp.
  11. Min-max hypersurface in manifold of positive Ricci curvature, J. Differential Geom. 105 (2017), no. 2, 291-343.
  12. On the free boundary min-max geodesics, Int. Math. Res. Not. IMRN 2016, no. 5, 1447-1466.
  13. 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.
  14. Mass angular momentum inequality for axisymmetric vacuum data with small trace, Comm. Anal. Geom. 22 (2014), no. 3, 519-571.
  15. Convexity of reduced energy and mass angular momentum inequalities (with R. Schoen), Ann. Henri Poincare 14 (2013), no. 7, 1747-1773.
  16. On the existence of min-max minimal surfaces of genus g≥ 2, Commun. Contemp. Math. 19 (2017), no. 4, 1750041, 36 pp.
  17. On the existence of min-max minimal torus, J. Geom. Anal. 20 (2010), no. 4, 1026-1055.

Research reports and surveys:

  1. Multiplicity One Conjecture in min-max theory, Partial Differential Equations, Oberwolfach Report 2019.
  2. On the Multiplicity One Conjecture in min-max theory, Surveys in Geometric Analysis 2019, to appear.
  3. Min-max theory for constant mean curvature (CMC) hypersurfaces (with Jonathan Zhu), Partial Differential Equations, Oberwolfach Report No. 35/2017.
  4. On minimal surfaces with free boundary (with Martin Li), special issues of ICCM Notices, to appear.
  5. 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 Seminars:

Past Conferences:

Past Teaching: