教师信息

陈学长

教授、博导

联系方式

Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing 210093, P.R. China

电子邮件

xuezhangchen@nju.edu.cn

研究领域

Geometric analysis and partial differential equations

办公地址

309-3 Meng Minwei Building, Math Department, Gulou Campus

主要课程

Spring 2024: Differential Geometry (undergraduate), Wednesday 9:00-12:00.

Fall 2023: Geometric Analysis, Wednesday 18:30-21:20.

Spring 2023: Differential Geometry (undergraduate), Wednesday 9:00-12:00.

Fall 2022: Geometric Analysis (Grandate and senior undergraduate students).

Fall 2021: Geometric Analysis, Wednesday 18:30-21:20.

Spring 2021: Differential Geometry (undergraduate), see textbook.

Fall 2020: Calculus, see here for 2017-2019 tests.

Spring 2020: Differential Geometry (undergraduate), see textbook.

Fall 2019: Partial Differential Equations (undergraduate), see textbook.

Spring 2019: Differential Geometry (undergraduate), see textbook.

Fall 2018: Partial Differential Equations (undergraduate), see textbook.

Spring 2018: Differential Geometry (undergraduate), see textbook.

Fall 2017: Geometric Analysis ( We mainly focus on the recent developments on the existence of a conformal metric with positive constant Q-curvature. The contents include the derivation of Q-curvature as well as Paneitz operator, the existence results when the related conformal invariance are positive, etc. This topic is selected from a seriece of papers by Sun-Yung Alice Chang-Paul Yang, M. Gursky, A. Malchiodi, F. B. Hang, Y. J. Lin et al. )

The following topic has been left to some future semester:

(textbook: Sorin Dragomir and Giuseppe Tomassini, Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, Vol 246, 2006. This couse is mainly concerned with the CR Yamabe problem and its resolution). Prerequisites of this class: Differential Geometry (Undergraduate) and something about Riemannian Geometry.

Spring 2017: (1) Differential Geometry (undergraduate), see textbook. (2) Discussion course (undergraduate, a little bit like a seminar): The compactness of the Yamabe problem.

Fall 2016: Geometric Analysis (plan to present relatively complete proofs of the existence of scalar-flat conformal metrics with constant boundary mean curvature on compact manifolds with boundary, including the formulation of this problem and motivation of the study on this problem).

Spring 2015: Partial Differential Equations II (the Yamabe problem and a brief introduction to the mean curvature problem on a compact manifold with boundary).

Spring 2014: Partial Differential Equations II (Geometric wave equations, focus on wave maps)

and Real Analysis, Calculus, etc.

个人简介

Born on 1st Aug. 1980, Chinese citizen; xuezhangchen@nju.edu.cn .

*Office Address:*

309-3 Meng Minwei Building, Math Department, Gulou Campus.

* Prerequisites of my future Phd students *

First of all, you should be interested in geometry, explicitly differential geometry and Riemannian geometry. Definitely you will benefit from the following two courses:

(i) Partial Differential Equations; (ii) Differential Geometry/ Riemannian Geometry.

Be patient and keep working hard. Please contact me directly by email with your CV and certificates/recommendations.

You are welcome to join Geometric Analysis Group at Nanjing University.

*Geometric Analysis and Differential Geometry S**eminar@NJU:*

Each is in general **two-hour** talk.

See here for invited lectures in Spring 2023, and the previous lectures: Fall 2019, Fall 2018, Spring 2018, Fall 2017.

*Working seminar：*

Fall 2020: Every Saturday 2 pm-5 pm, Room 1105 Minwei Meng Building.

Fall 2019: Every Saturday 2 pm-5 pm, Room 1105 Minwei Meng Building.

Spring 2019: Every Saturday 2 pm-5 pm, Room 1105 Minwei Meng Building.

Fall 2018: Every Saturday 2 pm-5 pm from March 17, Room 1105 Minwei Meng Building.

Spring 2018: Every Saturday 2 pm-5 pm from March 17, Room 1105 Minwei Meng Building.

Fall 2017: Every Saturday 2 pm-5 pm from September 19 2017, Room 1105 Minwei Meng Building.

Spring 2017: Every Saturday 2 pm-5 pm, Room 1105 Minwei Meng Building.

*Academic Visiting Experience and International Conferences:*

20th Feb. 2009 — 5th Apr. 2009 Visiting (Ph. D.) graduate student, Department of Mathematics, National University of Singapore, Singapore

5th Jan. 2011 — 8th Mar. 2011 Visiting scholar, Department of Mathematics, National University of Singapore, Singapore

19th Jan. 2012 — 18th Feb. 2012 Visiting scholar, Simons Center of Geometry and Physics, Stony Brook University, New York, USA

1st Apr. 2012 —30th April 2012 Visiting scholar, Department of Mathematics, National University of Singapore, Singapore

1st July 2012 —14th July 2012 Participant, AMSI/ANU/UQ Winter School, School of Mathematics and Physics, The University of Queensland Brisbane, Australia

21st Aug. 2013 — 19th Sep. 2013 Visiting scholar, Department of Mathematics, National University of Singapore, Singapore

8th Jan. 2014 — 16th Feb. 2014 Visiting scholar, Morningside Center of Mathematics, Chinese Academy of Sciences, China

28th July 2014 — 25th Aug. 2014 Visiting scholar, Department of Mathematics, National University of Singapore, Singapore

24th Nov. 2014 — 20th Dec. 2014 Visiting scholar, Program on scalar curvature in manifold topology and conformal geometry, Institute of Mathematical Sciences, National University of Singapore, Singapore

4th Sep. 2015 — 29th Aug. 2016 Visiting scholar, Department of Mathematics, Rutgers University, USA

20th Jan. 2017 — 18th Feb. 2017 Visiting scholar, Department of Mathematics, Rutgers University, USA

27th May 2017 — 3rd June 2017 Visiting scholar, Department of Mathematics, Sogang University, Korea

30th Sep. 2017 — 7th Oct. 2017 Visiting scholar, Department of Mathematics, Sogang University, Korea

24th Jan. 2018 — 14th Feb. 2018 Visiting scholar, Department of Mathematics, Rutgers University, USA

29th Apr. 2018 — 6th May 2018 Visiting scholar, Department of Mathematics, Sogang University, Korea

15th July 2018 — 11th Aug. 2018 Visiting scholar, Department of Mathematics, Princeton University, USA

26th Jan. 2019 — 1st Feb. 2019 Visiting scholar, Department of Mathematics, Rutgers University, USA

1st Feb. 2019 — 23th Feb. 2019 Visiting scholar, Department of Mathematics, Princeton University, USA

7th July 2019 — 14th July 2019 Visiting scholar, Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong

1st Aug. 2019 — 16th Aug. 2019 Visiting scholar, Department of Mathematics, Rutgers University, USA

2nd Oct. 2019 — 8th Oct. 2019 Visiting scholar, Department of Mathematics, Sogang University, Korea

31st Jan. 2020 — 23rd Feb. 2020 Visiting scholar, Department of Mathematics, Rutgers University, USA

17th June 2024—21st June 2024, The Ninth Pacific Rim Conference in Mathematics (PRCM)---Differential Geometry Section, Australian National University, Darwin Convention Centre, Northern Territory, Australia. See the poster.

教育经历

*Education:*

Sep. 2007—June 2011 Doctoral degree, Department of Mathematics & IMS, Nanjing University, P. R. China

Sep. 2002—July 2004 Master degree, Institute of Mathematical Science, Jilin University, P. R. China

Sep. 1998—July 2002 Undergraduate degree, School of Mathematical Science, Jilin University, P. R. China

工作经历

*Work Experience:*

July 2004—Oct. 2011 Assistant, Department of Applied Mathematics, Nanjing University of Science & Technology, P. R. China

Apr. 2011—Mar. 2013 Post-doctor, Beijing International Center for Mathematical Research, Peking University, P. R. China

Feb. 2012— Dec. 2018 Associate professor, Department of Mathematics, Nanjing University, P. R. China

Jan. 2019— present Professor, Department of Mathematics, Nanjing University, P. R. China

研究兴趣

Reserch interests: Geometric Analysis and Partial Differential Equations.

My recent research interests are rigidity theorems on compact manifolds with boundary; prescribed curvature problems and geometric flows in conformal geometry, such as, scalar curvature flow, Q-curvature flow, scalar curvature plus mean curvature flow, on compact manifolds; and fractional Q-curvature flow on the unit sphere.

学术兼职

学术奖励

100 National Excellent Doctoral Dissertations in China, Ministry of Education of China, 2013

New Century Excellent Talents in University, Ministry of Education of China, 2013

The 5th Jiangsu Award for Outstanding Achievements in Mathematics, Jiangsu Mathematical Society, 2017

Du Xia Award for Excellent in the Teaching of Mathematics for Undergraduates, Nanjing University, 2020

*Research and Travel Grants:*

Jan. 2023—Dec. 2026 Principal Investigator, Boundary conformal geometric problems and related geometric inequalities, *Natural Science Foundation of China **(No. 12271244)*

Jan. 2018—Dec. 2021 Principal Investigator, Constant scalar curvature and constant mean curvature problem and its related conformal geometric flows,* Natural Science Foundation of China (No. 11771204)*

July 2017—June 2020 Principal Investigator, the Start-up Grant of 2016 Deng Feng Program B, Nanjing University

Ky and Yu Fen Fan Fund Travel Grant from the AMS for the year 2015-2016, 2015

Awardee of Hwa Ying Foundation from Nanjing University for academic visit in the year 2015-2016, 2014

Jan. 2014—Dec. 2018 Principal Investigator, Nonlocal Q-curvature flow and the construction of type II blowup solutions to bi-harmonic heat equation with critical exponent, *Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 201417),* *Ministry of Education of China*

Jan. 2013—Dec. 2015 Principal Investigator, Partial differential equations in conformal geometry and liquid crystals, *Natural Science Foundation of China (No.* *11201223)*

Oct. 2011—Mar. 2013 Principal Investigator, Curvature flows in conformal geometry, *China Postdoctoral Science Foundation*

论文

*Preprints**:*

1. Xuezhang Chen and Pak Tung Ho, Q-curvature solitons, preprint (2015), 25 pp.

2. Xuezhang Chen and Pak Tung Ho, A fractional conformal curvature flow on the unit sphere, preprint (2019), 62 pp, arXiv:1906.08434.

3. Xuezhang Chen, Wei Wei and Nan Wu, Almost sharp Sobolev trace inequalities in the unit ball under constraints, preprint (2022), 73 pp, arXiv:2107.08647.

4. Xuezhang Chen and Wei Wei, The σ_{2}-curvature equation on a compact manifold with boundary, 103 pp, arXiv:2307.13942.

5. Xuezhang Chen and Shihong Zhang, Constrained Moser-Trudinger-Onofri inequality and a uniqueness criterion for the mean field equation, 22 pp, arXiv:2309.04706.

6. Xuezhang Chen and Yalong Shi, Green functions for GJMS operators on spheres, Gegenbauer polynomials and rigidity theorems, 36 pp, arXiv:2401.02087.

7. Xuezhang Chen and Shihong Zhang, A sharp Sobolev trace inequality of order four on three-balls, 29 pp, arXiv:2403.00380.

*Publications:*

1. Xuezhang Chen and Xingwang Xu, $Q$-curvature flow on the standard sphere of even dimension, J. Funct. Anal. 261 (2011), 934-980.

2. Xuezhang Chen and Xingwang Xu, The scalar curvature flow on $S^n$---perturbation theorem revisited, Invent. Math. 187 (2012), no. 2, 395-506; (with an erratum : Invent. Math. 187(2012), no. 2, 507-509).

3. Xuezhang Chen, Li Ma and Xingwang Xu, Remarks on the $Q$-curvature flow (in Chinese), Sci. Sin. Math. 44 (2014), 183-192. (An English version also appeared in arXiv:1312.5909).

4. Xuezhang Chen and Pak Tung Ho, Conformal curvature flows on a compact manifold of negative Yamabe constant, Indiana Univ. Math. J. 67 No. 2 (2018), 537–581.

5. Xuezhang Chen and Fei Hou, Remarks on GJMS operator of order six, Pacific J. Math. 289 (2017), no. 1, 35–70.

6. Xuezhang Chen, Pak Tung Ho and Liming Sun, Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant, Ann Global Anal Geom 53 (2018), 121–150; arXiv:1605.01719.

7. Xuezhang Chen and Liming Sun, Existence of conformal metrics with constant scalar curvature and constant boundary mean curvature on compact manifolds, *Commun. Contemp. Math.* 21 (2019), no. 3, 1850021, 51 pp; arXiv:1611.00229.

8. Xuezhang Chen, Mijia Lai and Fang Wang, Escobar-Yamabe compactifications for Poincare-Einstein manifolds and rigidity theorems, Adv. Math. 343 (2019), 16-35; arXiv:1712.02540.

9. Xuezhang Chen, Yuping Ruan and Liming Sun, The Han-Li conjecture in constant scalar curvature and constant boundary mean curvature problem on compact manifolds, Adv. Math. 358 (2019), 106854, 56 pp; arXiv:1805.09597.

10. Xuezhang Chen, Tianling Jin and Yuping Ruan, An existence theorem on the isoperimetric ratio over scalar-flat conformal classes, J. Differential Equations 269 (2020), no. 5, 4116–4136; arXiv:1910.01512.

11. Xuezhang Chen and Nan Wu, Blow-up phenomena for the constant scalar curvature and constant boundary mean curvature equation, J. Differential Equations 269 (2020), no. 11, 9432–9470; arXiv:1908.04815.

12. Xuezhang Chen, Mijia Lai and Fang Wang, The Obata equation with Robin boundary condition, Rev. Mat. Iberoam. 37 (2021), no. 2, 643–670; arXiv:1901.02206.

13. Xuezhang Chen, Mingxiang Li, Zirui Li and Xingwang Xu, On Gaussian curvature flow, J. Differential Equations 294 (2021), 178–250.

其他

*Organized Conferences and mini-courses:*

2017 Nanjing Workshop on Geometric Analysis for Younger Researchers, Nov. 25-26, 2017.

Mini-workshop on Geometric Analysis and Partial Differential Equations, June 23, 2018.

Winter Workshop on Geometric Analysis, Nov. 17-18, 2018.

Winter Workshop on Geometric Analysis, Nov. 30-Dec. 1, 2019.

Summer 2020 Online Talks on Partial Differential Equations @NJU: See here for online talks.

Mini-course by Professor Yanyan Li: The method of moving planes, September 23,25,28, 2020; here for lecture notes.

Summer Workshop on Geometric Analysis, June 24-27, 2021.

NJU-SJTU Joint Workshop on Geometric Analysis, Oct. 20-23, 2023.