个人简介
Brief CV
工作经历
- 2021.9 - 至今,副研究员- 中国科学院数学与系统科学研究院 ;
- 2018.9 - 2021.9,研究助理教授 - 香港中文大学;
- 2016.8 - 2018.8, 博士后研究员 - 加拿大麦吉尔大学, 合作导师: 管鹏飞教授 。
教育背景
- 2010.9 - 2016.6, 博士 - 基础数学, 中国科学技术大学, 导师: 麻希南教授 ;
- 2005.9 - 2009.6, 学士 - 数学与应用数学, 西北大学。
奖励和基金:
- 2019, 中国数学会钟家庆奖。
- 2021, 国家海外青年人才计划。
Employment
Education
- 2010.9 - 2016.6, Ph.D., University of Science and Technology of China. Advisor: Prof. Xinan Ma.
- 2005.9 - 2009.6, B.S. in Mathematics, Northwest University, China.
Honors and Funds:
- 2019, Zhong Jiaqing Mathematics Award.
- 2021, National Overseas Youth Talent Program.
研究方向: 偏微分方程,几何分析。
Research Area: Partial Differential Equations, Geometric Analysis.
主页: 数学所个人主页。
Website: on AMSS.
研究成果
Publications
代表作:
- 三维预定数量曲率超曲面的曲率内估计。
[摘要]
Amer. J. Math., 2024.
此文对三维正数量曲率方程的二阶导数内估计问题进行了完全解决。我们的贡献在于观察到一个具有有界平均曲率的等距嵌入,并通过得到平均值不等式,将点估计转化为相对容易的积分估计。在技术上,我们简化了一些步骤,在 Warren 和 袁域 的积分证明中避免使用 Sobolev 不等式。
- 三维欧式空间中sigma-2方程的Hessian内估计。
[摘要]
Front. Math , 2024.
此文不仅得到了三维 2-Hessian 方程的二阶导数内估计,还得到了两个重要引理。我们的贡献在于找到了一个量(log△u),并且在三维情形首次证明了它是强下调和函数。这一事实在 Shankar 和 袁域 的 2022 年 Duke 文章中得到了运用;另外一个是获得了双倍估计,这在 Shankar 和 袁域 最近的一篇文章 (arXiv:2305.12587) 中,解决四维 2-Hessian 方程内估计问题时发挥了关键作用。
- 预定数量曲率方程凸解的内C2正则性。
[摘要]
Duke Math. J., 2019. (合作者:管鹏飞)
此文得到了2-Hessian方程和预定数量曲率方程凸解的二阶内估计。我们的贡献在于首次引入了辅助函数 x ⋅ Du - u,并充分利用了 2-Hessian 算子的凹性和一阶导数条件,通过一个巧妙的极值原理得到了二阶内估计。不仅如此,我们还得到了一个几何应用。我们证明了具有正数量曲率的等距嵌入超曲面具有内部紧性,相信这将在研究等距嵌入等几何问题中有所应用。
- Hessian方程的Neumann问题。
[摘要]
Comm. Math. Phys., 2019. (合作者:麻希南)
此文完全解决了Trudinger 猜想。这边的主要难点是k‐Hessian 方程的解一般没有凸性,之前Lions—Trudinger—Urbas 在处理Monge—Amere 方程相应的边界估计的方法不再适用。通过精心构造精细的上下闸函数,我们建立了边界双法向导数估计。随后使用复杂的论证克服了这个困难。这种方法为解决更一般的全非线性偏微分方程的 Neumann 问题铺平了道路。
- Reilly公式的推广及其在Heintze-Karcher不等式中的应用。
[摘要]
Int. Math. Res. Not. IMRN, 2015. (合作者:夏超)
我们在空间形式中找到了Reilly积分公式的正确形式,这本质上是Bochner公式的积分版本。然后运用此Reilly公式,我们给出了Alexandrov刚性定理和Heintze–Karcher不等式在半球面和双曲空间中的新证明。该证明方法简单易懂,并已在许多论文中被使用。
我的文章:
Representative Works:
- Interior curvature estimates for hypersurfaces of prescribing scalar curvature in dimension three.
[abstract]
Amer. J. Math., 2024.
We prove a priori interior curvature estimates for hypersurfaces of prescribing scalar curvature equations in dimension three.
The new observation here is that we construct a Lagrangian graph
which is a submanifold of bounded mean curvature if the graph function of a hypersurface satisfies a scalar curvature equation.
Moreover, we apply a modified integral method of Warren-Yuan without relying on Sobolev inequality.
- Interior Hessian Estimates for sigma-2 Equations in Dimension Three.
[abstract]
Front. Math, 2024.
We prove a priori interior C2 estimates for 2-Hessian equations in dimension three.
The new ingredients in this paper are a trace Jacobi inequality and a doubling lemma which are crucial components
in the resolution of the four-dimensional case by Shankar-Yuan.
- Interior C2 regularity of convex solutions to prescribing scalar curvature equations, with Pengfei Guan.
[abstract]
Duke Math. J., 2019.
We provide a priori interior C2 estimates for scalar curvature equations in the convex case and prove compactness for higher-dimensional isometric embedding hypersurfaces. This interior estimate was motivated by the isometric embedding problem and was originally proven by E. Heinz in dimension 2. We successfully generalized Heinz's theorem to higher dimensions using a maximum principle argument. For the first time, a radial derivative x ⋅ Du - u is employed in this argument.
- The Neumann Problem for Hessian Equations, with Xi-Nan Ma.
[abstract]
Comm. Math. Phys., 2019.
We addressed Trudinger's Conjecture for the Neumann problem of Hessian equations. By carefully constructing delicate barrier functions, we established the boundary double normal derivative estimate. An intricate argument was then used to tackle the problematic terms.
The method paves the way for tackling the Neumann problem of more general fully nonlinear PDEs.
- A generalization of Reilly’s formula and its applications to a new Heintze-Karcher type inequality, with Chao Xia.
[abstract]
Int. Math. Res. Not. IMRN, 2015.
We prove a generalization of Reilly's formula in space forms, which is essentially an integral version of Bochner's formula. This generalized Reilly's formula is then applied to provide alternative proofs of Alexandrov's Theorem and the Heintze–Karcher inequality in both the hemisphere and hyperbolic space. The resulting proof is accessible and has been used and cited in many papers.
My papers:
科研活动
Activities
组织会议:
讨论班:
教学: @国科大
Conferences:
Seminars:
Teaching: @ UCAS
Copyright © 2024, Guohuan Qiu.