About Me
I was born in Zhejiang, China, and obtained my B.S. in Mathematics at Hangzhou Normal University (2022-2026), under the supervision of Professor Zhiyuan Xu.
Personal Awards
Curriculum Vitae
You can view and download my complete CV here: Qian Zhenye's CV (PDF)
Research
Research Interests
Partial Differential Equations and Geometric Analysis, including nonlinear elliptic PDEs, geometric measure theory.
Recently, I was interested in following topics:
- [November 2025 - now] Double-exponential growth of Euler equations in 2D, including the basic framework introduced by Kiselev-Sverak's work in 2014, called the small scale creation, however, recently, Andrej Zlatos, 6 July 2025, showed the desired growth on the half plane by a new idea, which extended the original method to unbounded domains.
- [October 2025 - now] Nonlinear instability of Navier-Stokes equations, including the linear stability established in Yudovich's book, the Bootstrap method established by Strauss-Guo in 1990s and its application in Friedlander-Pavlović-Shvydkoy's paper in 2006.
- [October 2025 - now] Convergence of minimal surfaces, including weak convergence and existence problem (see Colding-Minicozzi's book chapter 3-4), min-max method and its applications (see Colding-Minicozzi's book chapter 5 and Colding-De Lellis's survey in 2003). I recommend the beginners to study Fischer-Schoen's work in 1985 and Fischer-Schoen's work in 1990. On the other hand, B. White's note and Chodosh's note are also useful.
- [October 2025 - now] The regularity of elliptic partial differential equations, including Schauder estimate in Wang's paper in 2006, which can be extended to more general cases, such as the parabolic equations and fully nonlinear elliptic equations; However, I want to show the application on semilinear equations with certain convexity, which is inspired by Caffarelli-Friedman's work in 1985 and Caffarelli-Guan-Ma's work in 2006.
- [September 2025 - October 2025] Qualitative properties of semilinear elliptic partial differential equations, including the symmetry of semilinear equations, Liouville-type theorem of certain Schrödinger operator and energy estimate, motivated by Berestycki-Caffarelli-Nirenberg's work in 1997 and Alberti-Ambrosio-Cabré's work in 2001.
Preprints
Publications
Selected Talks
October 2025
A Short Review of Qualitative Properties for Semilinear Elliptic Equations
Elliptic Partial Differential Equations Seminar
Online
August 2024
The geometric topology of frontal surfaces
2024 International Workshop on Geometry, Topology of Singular Submanifolds and Related Topics
Northeast Normal University, Jilin, China
Study Experience
July 2025
USTC Institute of Geometry and Physics: Summer School in Geometry and Analysis
- Bernstein Problem for Minimal Surfaces and Beyond (Zhihan Wang, Cornell)
June 2025
ZJU: Partial Differential Equation Advanced Workshop
- Minmax Methods in Geometric Analysis (Tristan Rivière, ETH-Zurich)
August 2024
PKU BICMR: Summer School on Differential Geometry
- Riemannian Geometry (Chao Xia, XMU)
- Complex Geometry (Yalong Shi, NJU)
- Second order Elliptic Differential Equation (Jiakun Liu, UOW)
July 2024
USTC Institute of Geometry and Physics: Summer School in Geometry
- Some aspects of Ricci flow on non-compact manifolds (Eric Chen, UCB)
- Topics in mean curvature flow (Jinze Zhu, MIT)
- An introduction to decoupling inequalities in harmonic analysis (Shenwen Gan, WSCS)
Seminars
September 2025 - now
Elliptic Partial Differential Equations Seminar
Sunday 14:00-17:00, Room 101, Building 2, Hainayuan, Zhejiang University (School of Mathematical Sciences)
with H.J. Chen (ZJU), Z.J. Hu (ZJU), M.T. Zhu (WU), T.Y. Shen (ShanghaiTech), T.L. Wang (AMSS)
Selected videos: A Short Review of Qualitative Properties for Semilinear Elliptic Equations
June 2025 - now
Minimal Surfaces Seminar (online) - selected videos
The whole talks:
- A simple survey of minimal surfaces and related aspects 6.24 (Zhenye Qian, HZNU)
- The minimal surfaces equation and minimal submanifolds 7.5 (Haoyu Pan, HZNU)
- Applications of the first variation and Bernstein problem in two dimension 7.18 (Tian Zeng, CQU)
- The second Variation formula and Stability 8.1 (Tian Zeng, CQU)
- Bernstein problem I: The fundamental objects of Geometric Measure Theory 8.8 (Mutian Zhu, WU)
- Bernstein problem II: Regularity theory in Geometric Measure theory 8.15 (Mutian Zhu, WU)
- Bernstein problem II (conti): Regularity theory in Geometric Measure theory 8.22 (Mutian Zhu, WU)
- Bernstein problem III: Bernstein problem and Simons cone 8.28 (Mutian Zhu, WU)
- Small Energy Curvature Estimates for Minimal Surfaces 10.10 (Weipeng Ruan, XMU)
- Extension of Schoen-Simon-Yau via De Giorgi iteration (Bellettini 2025) 10.17 (Weipeng Ruan, XMU)
August 2024 - March 2025
Riemannian Geometry and Geometry Analysis, Hangzhou Normal University - selected videos
Selected topics I have talked:
- Riemannian Submanifold and Basic Equations, some aspects in Minimal Submanifolds in Euclidean space
- Comparison theorem in Riemannian Geometry (Rauch, Hessian and Laplacian, Volume)
- Cheeger-Gromov Convergence and Gromov's paracompactness theorem with some applications in lower bound Ricci curvature
- Bochner Technique on Riemannian Geometry and applications (Comparison theorems, Killing fields)
- Some aspects in CMC hypersurfaces: Alexandrov's theorem, Levy-Gromov Isoperimetric inequality
- Bernstein's theorem in Minimal surfaces and Plateau problem
August 2024 - December 2024
Algebraic Topology, Hangzhou Normal University
Selected topics I have talked:
- An Introduction to Cohomology Ring: Universal coefficient theorem, Cup product and Examples
- Some Applications of Myers-Vietors Argument: Künneth formula, Leray-Hirsch theorem in fiber bundle, Poincaré's duality for manifolds
- Some aspects in homotopy theory: Freudenthal suspension and introduction to stable homotopy groups, Hurewicz-Whitehead theorem and Motivation in Poincaré's Conjecture of 3-dim manifolds
Teaching
Teaching Assistant Experience
Spring 2025
Differential Geometry (Honor), Hangzhou Normal University
Instructor: Zhiyuan Xu
References: exercises and supplementary materials
Autumn 2024
Analytic Geometry, Hangzhou Normal University
Instructor: Zhiyuan Xu
References: exercises and supplementary materials
Notes & Materials
Lecture Notes
Miscellaneous
Recent Thoughts
October 2025
I listened an interesting talk by professor Xiao Zhong about the Dimer models and Beltrami equation from his seminal work in 2025 joint with Astala, Duse and Prause.
A simple model of Random tilings is that, one may tile a hexagon (with unit edges) with different lozenges (3 different orientations, denoted by different colors, say yellow, red and blue) with side length $1/n$, which can be equivalently regarded as cutting a cube (3D) to a steped surface (view from $(1,1,1)$-direction). We express this surface by a height function, say $h$, which is piecewice linear (Lipschitz continuous) and $\nabla h\in \{(0,\sqrt{3}),\ (1,0),\ (-1,0)\}$. A natural question is that,
Indeed, the question is related to an interesting theorem (conjecture), called The Arctic Circle Theorem, established by Jockusch-Propp-Shor in 1994 and Cohn-Larsen-Propp in 1998.
Why we guess that there exists a sure surface as the limitation of certain convergence? Indeed, the conjecture was motivated by some necessary tiling method - for example, if you want to tile the bottom angle of hexagon, the probabilities of different choice are different for the sake of the orientation of angle and lozenges. Hence, we expect certain geometric and combinational shape of limited surface.
We present the limit shape theorem, by several seminal works by Chon-Kenyon-Propp-Okounkov-Sheffield, namely,
On the other hand, the convergence problem is related to minimizers of certain variational. Indeed, the asymptotic height function $h(z)$ minimizes \[ \inf\left\{\int_\Omega \sigma(\nabla v):\ v\ \text{is Lipschitz,}\ \nabla v\in N,\ v=h_0\ \text{on}\ \partial \Omega\right\} \] where the surface tension $\sigma$ is convex in a convex polygon $N$ and satisfies the following Monge–Ampère equation, \[ \begin{cases} \mathrm{det} D^2\sigma=1,& \text{in}\ N\\ \sigma=0,& \text{on}\ \partial N \end{cases} \]
Additionally, De Silva and Savin showed the $C^1$-regularity of the minimizer of functional above in 2D, see their paper Minimizers of convex functionals arising in random surfaces.
Given a linear elliptic operator $L:=\sum a_{ij}(x)\partial_{ij}+\sum_ib_i(x)\partial_i+c(x)$ - where we omit certain (uniform) elliptical properties - and a bounded domain $\Omega\subset\mathbb{R}^n$, we say that the maximum principle, simply say m.p. holds for $L$ in $\Omega$, if \[ \begin{cases} Lz\ge 0,& \text{in}\ \Omega\\ \limsup_{x\to \partial\Omega}z\le 0,& \text{on}\ \partial\Omega \end{cases}\Longrightarrow z\le 0\ \text{in}\ \Omega,\quad \text{where $z$ is a function on $\Omega$} \] A simple philosophy of the m.p. is that the boundary value dominates the supremum of the interior value, which is equivalent to a certain comparison principle, say c.p., however, we regard it as m.p. in this version; for example, let $z=u-v$, where $u,v$ are two given functions to be compared. Hence, the m.p. may be used to show certain monotonicity relation.
Unfortunately, we have known that there are three sufficient conditions of m.p. for $L$, see Berestycki-Nirenberg's statement in 1991 (also in Han-Lin's note in 2011),
- i) "The free term" $c(x)\le 0$. See any elliptic PDEs book, such as Gilbarg-Trudinger's well-known book in 1978.
- ii) If we can find a positive function $g$, such that $Lg\le 0$, then there exists a new elliptic operator, say $\tilde{L}$ with nonpositive free term, such that $\tilde{L}(z/g)\ge 0$.
- iii) Roughly, if the domain $\Omega$ is "narrow", for instance $\Omega=(0,1)\times(0,\epsilon)$, where $\epsilon\ll 1$. Indeed, we can construct $g(x_1)$ satisfies ii) in this version.