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:

Differential Geometry, Calculus of Variations, General Relativity.

Preprints:

On the Multiplicity One Conjecture in min-max theory, arXiv:1901.01173.
Min-max theory for networks of constant geodesic curvature (with Jonathan Zhu), arXiv:1811.04070.
Existence of hypersurfaces with prescribed mean curvature I - Generic min-max (with Jonathan Zhu), arXiv:1808.03527.
Min-max minimal disks with free boundary in Riemannian manifolds (with Longzhi Lin and Ao Sun), arXiv:1806.04664.
Compactness and generic finiteness for free boundary minimal hypersurfaces (with Qiang Guang), arXiv:1803.01509.
Free boundary minimal hypersurfaces with least area (with Qiang Guang and Zhichao Wang), arXiv:1801.07036.

Publications:

Min-max theory for CMC hypersurfaces (with Jonathan Zhu), accepted by Invent. Math, arXiv:1707.08012.
Min-max theory for free boundary minimal hypersurfaces I - regularity theory (with Martin Li), accepted by J. Differential Geometry, arXiv:1611.02612.
A maximum principle for free boundary minimal varieties of arbitrary codimension (with Martin Li), accepted by Comm. Anal. Geom., arXiv:1708.05001.
Curvature estimates for stable free boundary minimal hypersurfaces (with Qiang Guang and Martin Li), accepted by J. Reine Angew. Math (Crelle's Journal), DOI: https://doi.org/10.1515/crelle-2018-0008, arXiv:1611.02605.
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 .
Entropy of closed surfaces and min-max theory (with D. Ketover), J. Differential Geom. 110 (2018), no. 1, 31-71.
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.
Min-max hypersurface in manifold of positive Ricci curvature, J. Differential Geom. 105 (2017), no. 2, 291-343.
On the free boundary min-max geodesics, Int. Math. Res. Not. IMRN 2016, no. 5, 1447-1466.
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.
Mass angular momentum inequality for axisymmetric vacuum data with small trace, Comm. Anal. Geom. 22 (2014), no. 3, 519-571.
Convexity of reduced energy and mass angular momentum inequalities (with R. Schoen), Ann. Henri Poincare 14 (2013), no. 7, 1747-1773.
On the existence of min-max minimal surfaces of genus g≥ 2, Commun. Contemp. Math. 19 (2017), no. 4, 1750041, 36 pp.
On the existence of min-max minimal torus, J. Geom. Anal. 20 (2010), no. 4, 1026-1055.

Research reports and surveys:

Min-max theory for constant mean curvature (CMC) hypersurfaces (with Jonathan Zhu), Partial Differential Equations, Oberwolfach Report No. 35/2017.
On minimal surfaces with free boundary (with Martin Li), special issues of ICCM Notices, to appear.
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:

Lectures on minimal surfaces: This series of lecture notes were taken by Junrong Yan for the topic course I taught during Fall 2017.
Introduction to the min-max theory for minimal surfaces: Hand-written lecture notes for a topic course on the min-max theory of minimal surfaces in 2013.
Lecture notes on minimal surfaces: This series of lecture notes were taken for the topic course on minimal surfaces given by Professor Rick Schoen in the Winter quarter of 2012 at Stanford.
Introduction to Mathematical General Relativity: This series of lecture notes were taken for the topic course on mathematical General Relativity given by Professor Rick Schoen in the spring quarter of 2012 at Tsinghua University.

Slides and videos:

Multiplicity One Conjecture in min-max theory (Part I) (Part II): This is the video of the lectures given in the Variational Methods in Geometry Seminar, IAS, March 2019.
Min-max theory for constant mean curvature (CMC) hypersurfaces: This is the video of the lecture given in the Workshop: Mass in General Relativity at Simons Center, Stony Brook, March 2018.
Min-max minimal hypersurface with free boundary: This is the video of the lecture given in the Geometric Analysis and General Relativity workshop at BIRS, Banff, July 2016.

Past Conferences:

Past Teaching:

Fall 2016. Math 240A: Lectures on Differential Geometry.
Winter 2017. Math 240B: Riemannian Geometry and Math 117: Methods of Analysis.
Fall 2017. Math 241A: Topics in Differential Geometry, Math 117: Methods of Analysis.
Spring 2018. Math 8: Transition to higher mathematics.