https://neckpinch.com/authors/maxwell-stolarski/New arXiv papers by Maxwell Stolarski, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2607.03152arxiv:2607.03152Fri, 03 Jul 2026 00:00:00 GMTMaxwell Stolarski, Wei-Bo Su — For each integer K≥2 when n≥4, and for K=2,3,4 when n=3, we construct an almost-calibrated Lagrangian mean curvature flow L_K(t) in ℂⁿ, 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) |A_L_K(t)| ∼ (T-t)^(−K/2) as t↗ 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.Mean curvature flowLagrangian mean curvature flowSingularitiesHomogeneous spacesEigenvalues & spectrum https://arxiv.org/abs/2606.21541arxiv:2606.21541Fri, 19 Jun 2026 00:00:00 GMTMaxwell Stolarski, Wei-Bo Su — Let 𝓒 be a G-invariant special Lagrangian cone admitting a scaled family of G-invariant special Lagrangian desingularizations a overline L which converge to 𝓒 as a↘ 0. We study the linearized self-shrinker operator on aoverline L in a Gaussian weighted L² space of G-equivariant functions. For 0<a≪1, 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.Mean curvature flowLagrangian mean curvature flowSelf-shrinkers & translatorsSingularitiesEigenvalues & spectrum https://arxiv.org/abs/2604.15198arxiv:2604.15198Thu, 16 Apr 2026 00:00:00 GMTMaxwell Stolarski, Alex Waldron — 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.Ricci flowRicci-flat & Calabi–YauStability & uniquenessHarmonic maps & DeTurck trick https://arxiv.org/abs/2311.16262arxiv:2311.16262Mon, 27 Nov 2023 00:00:00 GMTMaxwell Stolarski — This paper studies singularities of mean curvature flows with integral mean curvature bounds H ∈ L^∞ L^pₗoc for some p ∈ ( n, ∞]. For such flows, any tangent flow is given by the flow of a stationary cone C. When p = ∞ and 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 ⊆ ℝ^N with H ∈ L^∞ L^pₗoc. For smooth, codimension one mean curvature flows with H ∈ L^∞ L^∞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.Mean curvature flowSingularitiesWeak & singular flowsMinimal & CMC surfaces https://arxiv.org/abs/2202.03386arxiv:2202.03386Mon, 07 Feb 2022 00:00:00 GMTMaxwell Stolarski — 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.Ricci flowRicci solitonsSingularitiesKähler geometryBoundaries & non-compact