arXiv subjects

Maxwell Stolarski

Publications and source records attributed to Maxwell Stolarski.

8 recordsLinked to original sources

arXiv2026

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

For each integer $K\geq2$ when $n\geq4$, and for $K=2,3,4$ when $n=3$, we construct an almost-calibrated Lagrangian mean curvature flow $L_K(t)$ in $\mathbb{C}^{n}$, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time $T$ with the explicit curvature blow up rate \[ \sup_{L_{K}(t)} |\mathbf{A}_{L_{K}(t)}| \sim (T-t)^{-K/2} \qquad \text{as } t\nearrow T . \] The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.

math.DG↗

arXiv2026

Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

Let $\mathcal C$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a \overline L$ which converge to $\mathcal C$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline L$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll1$, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.

math.DG↗

arXiv2026

Integrable Deformations and Stability of the Ricci Flow

We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the $L^p$-stability theorems of Deruelle-Kroncke and Kroncke-Petersen.

math.DG↗

arXiv2023

On the Structure of Singularities of Weak Mean Curvature Flows with Mean Curvature Bounds

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$. When $p = \infty$ and $\mathbf{C}$ is a regular cone, we prove that the tangent flow is unique. These results hold for general integral Brakke flows of arbitrary codimension in an open subset $U \subseteq \mathbb{R}^N$ with $H \in L^\infty L^p_{loc}$. For smooth, codimension one mean curvature flows with $H \in L^\infty L^\infty_{loc}$, we also show that, at points where a tangent flow is given by an area-minimizing Simons cone, there is an accompanying limit flow given by a smooth Hardt-Simon minimal surface.

math.DG↗

arXiv2022

Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers

Given an asymptotically conical, shrinking, gradient Ricci soliton, we show that there exists a Ricci flow solution on a closed manifold that forms a finite-time singularity modeled on the given soliton. No symmetry or Kahler assumptions on the soliton are required. The proof provides a precise asymptotic description of the singularity formation.

math.DG↗

arXiv2020

Existence of Mean Curvature Flow Singularities with Bounded Mean Curvature

In [Vel94], Velazquez constructed a countable collection of mean curvature flow solutions in $\mathbb{R}^N$ in every dimension $N \ge 8$. Each of these solutions becomes singular in finite time at which time the second fundamental form blows up. In contrast, we confirm here that, in every dimension $N \ge 8$, a nontrivial subset of these solutions has uniformly bounded mean curvature.

math.DG↗

arXiv2019

Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.

math.DG↗

arXiv2017

Steady Ricci Solitons on Complex Line Bundles

We show the existence and uniqueness of a one-parameter family of smooth complete $U(1)$-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano Kähler-Einstein base with first Chern class proportional to that of the base. These solitons are non-Kähler except on the total space of the canonical bundle.

math.DG↗