2–9 October 2026 · Hessian equations and geometric flows
This issue follows two themes: obtaining second-derivative estimates from the structure of a nonlinear equation, and extracting geometric information from a flow even when smoothness or uniform bounds break down. The four selected papers were first submitted to arXiv during this reporting period.
Checked on 9 October 2026. Dates below are arXiv submission dates (UTC). These are summaries of preprint statements and proof strategies, not independent verification of every proof.
1. Hessian quotient estimates through compactness
Jianxiang Liu · 6 October · New proof of existing estimates
Can interior second derivatives be controlled without deriving a lengthy Jacobi inequality? Liu gives compactness proofs for smooth convex solutions of
\[\frac{\sigma_n(D^2u)}{\sigma_{n-2}(D^2u)}=1, \qquad \frac{\sigma_n(D^2u)}{\sigma_{n-1}(D^2u)}=f(x,u,Du)>0.\]Here $\sigma_j$ is the $j$th elementary symmetric polynomial. In the second equation, $f$ is $C^2$ and $1/f$ must be concave in the gradient variable. The estimates depend on the stated lower-order bounds and data.
Method and significance. A modified Legendre transform turns a hypothetical blow-up sequence into a compactness problem. A strong minimum principle and the area formula then yield a contradiction. This revisits estimates of Lu and Fung; the contribution is a different proof mechanism. For Hessian regularity, it offers a useful alternative to Jacobi-inequality calculations. Convexity and the condition on $1/f$ remain essential hypotheses of the stated results.
Source: Hessian Estimates for Convex Solutions of Hessian Quotient Equations via Compactness, Theorems 1.1–1.2 and Sections 2–3. Read the paper.
2. Oblique boundary problems without domain convexity
Zhibo Hu and Feida Jiang · 8 October · Global estimates and solvability
The authors study equations of the form
\[F\!\left(\gamma\Delta u\,I-D^2u-A(x,u,Du)\right)=B(x,u,Du), \qquad \gamma>1,\]with oblique boundary conditions. Their estimates dispense with convexity of the domain and the usual strict regularity condition on $A$ in the gradient variables.
What is established? Under the paper’s structural and growth assumptions, they derive global gradient and second-derivative estimates. The classical existence and uniqueness theorem concerns semilinear oblique data and also assumes admissible sub- and supersolutions, monotonicity and the additional hypotheses of Theorem 1.4.
Method and significance. The uniformly elliptic structure supplied by $\gamma>1$ supports interior estimates and boundary barriers. This connects Hessian techniques with nonlinear oblique boundary theory and conformal geometry. The restriction $\gamma>1$ matters: the endpoint $\gamma=1$ is not covered by this argument, and this is not the ordinary $\sigma_{n-1}(D^2u)$ equation.
Source: Oblique boundary value problems for n-1type augmented Hessian equations, Theorems 1.1–1.4 and Remark 1.3. Read the paper.
3. Which hypersurface survives longest under mean curvature flow?
Beomjun Choi, Wenkui Du, Seung Chul Park and Junseo Youn · 7 October · Sharp geometric inequality
For a fixed initial hypersurface area, the authors establish that a shrinking round sphere maximizes extinction time. For a smooth bounded domain $\Omega\subset\mathbb R^{n+1}$, their bound for its level-set flow and the associated integral Brakke flows is
\[T_{\mathrm{ext}}\leq \frac{1}{2n} \left(\frac{P(\Omega)}{|\mathbb S^n|}\right)^{2/n},\]where $P(\Omega)$ is boundary area. Equality forces a round ball; for the Brakke statement the equality flow is the standard multiplicity-one shrinking sphere.
Method and significance. The proof combines outward-minimizing hulls, smooth approximations, Ilmanen’s elliptic regularization and Minkowski inequalities on almost every time slice. The central difficulty is retaining a sharp inequality through singular times. This links convex-geometric inequalities with weak geometric evolution. Quantitative stability and extensions to other ambient settings remain questions raised by the authors, rather than consequences already proved here.
Source: Sharp dynamical isoperimetric principle for mean curvature flow, Theorem 1.1, the introduction’s proof strategy and Remark 6.5. Read the paper.
4. A Robin boundary condition produces curvature concentration
Tianhong Pu, Xiaoqian Xin, Lixia Yuan and Xiaoyu Zhang · 6 October · Long-time asymptotics
The paper studies positive even graphical solutions of forced planar curvature flow $V=H+b$, $b>0$. On the half-strip, the boundary conditions are $u_x(0,t)=0$ and $u_x(1,t)=u(1,t)$, with smooth compatible initial data; the normalized-slope results additionally assume $0<u_0^{\prime}/u_0<1$ in $(0,1)$.
What is established? The authors report global existence and divergence of height and velocity. Away from the symmetry axis, the slope diverges but $u_x/u\to1$ and $u_t/u\to b$. Curvature tends to zero in $L^1([a,1])$ for every $a>0$, while the horizontal measure $H\,dx$ converges to a unit atom at the axis. This measure is distinct from curvature weighted by arclength.
Method and significance. Translating comparison profiles, the one-dimensional zero-number principle and normalized variables reveal a limit despite unbounded gradients. The result concerns this planar Robin problem; it does not assert convergence for general higher-dimensional curvature flows.
Source: Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions, Theorem 1.1 and Sections 3–6. Read the paper.