Xin Zhou
Professor
Address: Department of Mathematics
531 Malott Hall
Ithaca, NY 14853
E-mail: xinzhou@cornell.edu
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Research Interest:
Research Interest:
Differential Geometry and Calculus of Variations.
Awards and Honors:
Awards and Honors:
Fellow of the American Mathematical Society for 2024.
Von Neumann Fellowship, IAS, 2024.
Simons Fellows in Mathematics, 2023-2024.
Invited speaker, ICM 2022.
NSF Career Award, DMS-2243149, DMS-1945178, 2020-2024.
Alfred P. Sloan Research Fellow, 2019-2021.
NSF Grant DMS-1811293, 2018-2021.
NSF Grant DMS-1704393, DMS-1406337, 2014-2017.
Preprints:
Preprints:
An alternative for constant mean curvature hypersurfaces (with L. Mazurowski), arXiv:2408.13864.
Infinitely many half-volume constant mean curvature hypersurfaces via min-max theory (with L. Mazurowski), arXiv:2405.00595.
Existence of four minimal spheres in $S^3$ with a bumpy metric (with Z. Wang), arXiv:2305.08755.
Publications:
Publications:
Improved $C^{1,1}$ regularity for multiple membranes problem (with Z. Wang), Peking Math J (2025), arXiv:2308.00172.
The half-volume spectrum of a manifold (with L. Mazurowski), Calc. Var. 64, 155 (2025), arXiv:2302.07722.
Min-max minimal hypersurfaces with higher multiplicity (with Z. Wang), J. Differential Geom. 129(1): 225-260 (January 2025), arXiv:2201.06154.
Min-max theory for capillary surfaces (with C. Li and J. Zhu), J. reine angew. Math. 818 (2025), 215-262, arXiv:2111.09924.
Multiplicity one for min-max theory in compact manifolds with boundary and its applications (with A. Sun and Z. Wang), Calc. Var. 63, 70 (2024), arXiv:2011.04136.
Existence of curves with constant geodesic curvature in a Riemannian 2-sphere (with D. Cheng), Trans. Amer. Math. Soc., 374 (2021), 9007-9028, arXiv:2106.12374.
Existence of constant mean curvature 2-spheres in Riemannian 3-spheres (with D. Cheng), Comm. Pure Appl. Math. 76 (11), 3374-3436 (2023), arXiv:2012.13379.
Generic scarring for minimal hypersurfaces along stable hypersurfaces (with A. Song), Geom. Funct. Anal. 31, 948-980 (2021), arXiv:2006.03038.
Min-max theory for free boundary minimal hypersurfaces II -- General Morse index bounds and applications (with Q. Guang, M. Li, and Z. Wang), Math. Ann. 379, 1395–1424(2021), arXiv:1907.12064.
On the Multiplicity One Conjecture in min-max theory, Ann. of Math. (2) 192 (2020), no. 3, 767–820, arXiv:1901.01173.
Min-max theory for networks of constant geodesic curvature (with J. Zhu), Adv. Math. 361 (2020), 106941, 16 pp.
Existence of hypersurfaces with prescribed mean curvature I - Generic min-max (with J. Zhu), Camb. J. Math. 8 (2020), no. 2, 311-362, arXiv:1808.03527.
Min-max minimal disks with free boundary in Riemannian manifolds (with L. Lin and A. Sun), Geom. Topol. 24 (2020) 471-532.
Compactness and generic finiteness for free boundary minimal hypersurfaces (I) (with Q. Guang and Z. Wang), Pacific J. Math. Vol. 310 (2021), No. 1, 85–114, arXiv:1803.01509.
Free boundary minimal hypersurfaces with least area (with Q. Guang and Z. Wang), Comm. Anal. Geom. pp. 1177-1215, Volume 31 (2023) Number 5, arXiv:1801.07036.
Min-max theory for constant mean curvature hypersurfaces (with J. Zhu), Invent. Math. 218 (2019), no. 2, 441-490, arXiv:1707.08012.
Min-max theory for free boundary minimal hypersurfaces I - Regularity theory (with M. Li), J. Differential Geom.118 (2021), no. 3, 487-553, arXiv:1611.02612.
A maximum principle for free boundary minimal varieties of arbitrary codimension (with M. Li), Comm. Anal. Geom., 29 (2021), No. 6, 1509-1521, arXiv:1708.05001.
Curvature estimates for stable free boundary minimal hypersurfaces (with Q. Guang and M. Li), J. reine angew. Math. 759 (2020), 245-264.
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. 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 H. 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. 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:
Research Reports and Surveys:
Recent progress on Geometric Variational Theory (with Tongrui Wang, in Chinese), SCIENTIA SINICA Mathematica, 2023, 53(10): 1287–1302.
Mean curvature and variational theory, Proc. Int. Cong. Math. 2022, Vol. 4, pp. 2696–2717.
Morse theory for minimal hypersurfaces and the Multiplicity One Conjecture (with Zhichao Wang), Surveys in Geometric Analysis 2019, 189-203, Science Press Beijing, Beijing, 2020. ISBN: 9787030654427.
Multiplicity One Conjecture in min-max theory, Partial Differential Equations, Oberwolfach Report, No. 34/2019.
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), a survey article, preprint.
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:
Lecture Notes:
Iteration methods in PDE and application in geometry: A set of lecture notes on iteration methods in PDE and applications in geometry that I taught at Cornell in 2024. The notes include the De Girogi-Nash-Moser theory in PDE, epsilon regularity of harmonic maps, Allard regularity for stationary varifolds, and regularity for stable minimal hypersurfaces.
Lectures on min-max theory of minimal hypersurfaces: A set of lecture notes on the min-max theory of minimal hypersurfaces that I taught at Cornell in 2022. The notes were taken by Xingzhe Li.
Lectures on minimal surfaces: A set of lecture notes on minimal surfaces that I taught at UCSB in 2017 and at Cornell in 2021.
Lecture notes on minimal surfaces: This series of lecture notes were taken for the topic course on minimal surfaces taught by Professor Rick Schoen at Stanford in 2012.
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.
Talk Videos:
Talk Videos:
Half-volume spectrum and CMC hypersurfaces. IMPA, March 2025.
Existence of four minimal spheres in S^3 with a bumpy metric. IAS, Feb. 2024.
Some recent development in minimal surface theory. The inaugural International Congress of Basic Science, Beijing, July 2023.
Recent development of constant mean curvature hypersurfaces. Not Only Scalar Curvature Seminar, Feb. 2023.
Mean curvature and variational theory: ICM invited talk, July 2022.
Min-max minimal hypersurfaces with higher multiplicity: MSRI meeting, March 2022.
Multiplicity One Conjecture in min-max theory (Part I) (Part II): Variational Methods in Geometry Seminar, IAS, March 2019.
Min-max theory for constant mean curvature (CMC) hypersurfaces: Workshop: Mass in General Relativity at Simons Center, Stony Brook, March 2018.
Min-max minimal hypersurface with free boundary: Geometric Analysis and General Relativity workshop at BIRS, Banff, July 2016.
Past Conferences:
Past Conferences:
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