Differential Geometry Papers of the Week

However, existing reviews usually organise the literature by algorithm category or application scenario, making it difficult to explain both the behavior ultimately produced by the system and the logic through which the controller achieves its objectives. This scoping review aims to map the distribution and thematic structure of the literature on reinforcement learning-based control of nonlinear dynamical systems, from 2021 to 2026.

We study the Riemannian geometry of the deep linear network as a foundation for a thermodynamic description of the learning process.

We investigate Bernstein theorem for the affine maximal type equation \begin{equation*} \sum_{i,j=1}^nf^{ij}(D^a)_{ij}=0,\qquad D=\det(f_{ij}),\qquad a\neq0, \end{equation*} under completeness of the Calabi metric.

We focus primarily on the strong condition that the associated torsion $3$-form is closed.

In this perspective, we highlight recent advances in electrochemical skeletal editing until September 2026, with particular emphasis on electrochemical atom insertion, ring expansion, and ring contraction strategies.

For a fixed constant $\alpha>0$ and $2\le k\le n-1$, we study the prescribed sum Hessian curvature equation \[ \sigma_k(\lambda(\eta))+\alpha\sigma_{k-1}(\lambda(\eta))=f(X,\nu).

We establish uniqueness of Ricci flows with non-negative sectional curvatures and scaling invariant curvature bounds, assuming that the induced distance functions converge to the same initial distance function up to isometry.

We study the equality case of the inequalities between the Lyapunov spectrum and the Harder--Narasimhan slopes of a semi-simple local system with non-expanding cusp monodromy over a finite-area hyperbolic surface.

We prove that $F$ is Berwald if, at each fixed base point, the Minkowski norm is real-analytic on the entire slit tangent space.

We give a proof of a folklore conjecture that a compact Kähler manifold with the same scalar Laplace spectrum as complex projective space equipped with the Fubini--Study metric is holomorphically isometric to it.

We study Hodge--Riemann relations for ordinary, sequential, and weak $m$-positive real $(1,1)$-forms in complex dimension $d$.

For $n\geq2$ and $\theta\in(0,\pi/2)$, we study the long-time existence and asymptotic behavior of an even normalized capillary $L_p$ dual Minkowski flow in the Euclidean half-space $\overline{\mathbb{R}^{n+1}_+}$.

In this paper, we study the anisotropic mean curvature flow in the Gaussian space endowed with a Finsler metric obtained from a Minkowski metric by a conformal deformation.

On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern--Calabi flow with a finite-time Chern Type-I singularity at any prescribed point.

We establish a logarithmic Sobolev inequality for complete $C^{1,1}$ immersed submanifolds of dimension $m\ge2$ in Cartan--Hadamard manifolds with sectional curvature at most $-b^2$.

In this paper, we prove that we can subtract the full spectral gap $\left(\frac{N-1}{p}\right)^{p}$ of the $p$-Laplacian and keep the sharp Sobolev constant $S(N,p)$ of $\mathbb{R}^N$ at the same time in a large family of $p$.

In this paper, we identify a region of prescribed combinatorial curvature vectors, allowing some negative components, for which existence still holds.

We study holomorphic sectional curvature and the Calabi curvature operator on compact Kähler manifolds.