Xin Zhou

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

Research Interest:

• Differential Geometry, Calculus of Variations, General Relativity.

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), International Mathematics Research Notices, Volume 2018, Issue 4, 20 .
3. Entropy of closed surfaces and min-max theory (with D. Ketover), accepted by J. Differential Geometry, arXiv:1509.06238.
4. Existence of minimal surfaces of arbitrarily large Morse index (with Haozhao. Li), Calculus of Variations and Partial Differential Equations, 2016, 55(3), 1-12.
5. Min-max hypersurface in manifold of positive Ricci curvature, J. Differential Geometry, 105 (2017), 291-343.
6. On the free boundary min-max geodesics, International Mathematics Research Notices Vol. 2016, No. 5, pp. 1447-1466.
7. Min-max minimal hypersurface in $(M^{n+1}, g)$ with $Ric_{g}>0$ and $2\leq n\leq 6$, J. Differential Geometry, 100 (2015) 129-160.
8. Mass angular momentum inequality for axisymmetric vacuum data with small trace, Communication in Analysis and Geometry, 22, (2014) 519-571.
9. Convexity of reduced energy and mass angular momentum inequalities (with R. Schoen), Ann. Henri Poincar\'e, 14 (2013), 1747-1773.
10. On the existence of min-max minimal surfaces of genus g≥ 2, Commun. Contemp. Math., 19, 1750041 (2017).
11. On the existence of min-max minimal torus, J. Geom. Anal. 20 (2010), 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), Proceeding of the Workshop on Geometric Analysis 2017, to appear.

Other writings:

• Introduction to the min-max theory for minimal surfaces: Hand-written lecture notes for a topic class 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 class 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 class on mathematical General Relativity given by Professor Rick Schoen in the spring quarter of 2012 at Tsinghua University.

Slides and videos:

• 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.
• Geometric variational theory and applications: This is the job talk slides given during Fall 2015.

Past Conferences:

• Geometry and Analysis on Manifolds 2017.
• Geometry and Analysis on Manifolds 2018.

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.