https://neckpinch.com/authors/william-p-minicozzi/New arXiv papers by William P. Minicozzi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2605.10589arxiv:2605.10589Mon, 11 May 2026 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We will show that the distance between two minimal hypersurfaces is a Lipschitz continuous supersolution, in the viscosity sense, of a natural elliptic partial differential equation. This not only recovers several well-known properties of minimal hypersurfaces, but also encodes substantially richer information. Moreover, if the reference hypersurface is allowed to evolve by mean curvature flow, one obtains comparably strong estimates for a corresponding parabolic PDE, leading in particular to local Harnack inequalities for the distance. There is even a fully parabolic extension in which both hypersurfaces evolve. The problem of tracking the distance between two evolving hypersurfaces arises naturally in a wide range of settings.Mean curvature flowHarnack inequalitiesMinimal & CMC surfaces https://arxiv.org/abs/2502.03634arxiv:2502.03634Wed, 05 Feb 2025 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We show how to use the arguments of [CM2] to get a stronger effective version of uniqueness of blowups that has a number of consequences.Mean curvature flowSingularitiesStability & uniqueness https://arxiv.org/abs/2306.08078arxiv:2306.08078Tue, 13 Jun 2023 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.Mean curvature flowRicci solitons https://arxiv.org/abs/2305.03135arxiv:2305.03135Thu, 04 May 2023 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates outwards for shrinking solitons.Ricci solitons https://arxiv.org/abs/2305.02416arxiv:2305.02416Wed, 03 May 2023 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.Ricci flowEigenvalues & spectrum https://arxiv.org/abs/2202.09839arxiv:2202.09839Sun, 20 Feb 2022 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle, especially for non-compact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism.Ricci flowSingularitiesBoundaries & non-compact https://arxiv.org/abs/2109.06240arxiv:2109.06240Mon, 13 Sep 2021 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — Comparing and recognizing metrics can be extraordinarily difficult because of the group of diffeomorphisms. Two metrics, that could even be the same, could look completely different in different coordinates. This is the gauge problem. The general gauge problem is extremely subtle for non-compact spaces. Often it can be avoided if one uses some additional structure of the particular situation. However, in many problems there is no additional structure. Instead we solve the gauge problem directly in great generality. The techniques and ideas apply to many problems. We use them to solve a well-known open problem in Ricci flow: Strong rigidity of cylinders. Strong rigidity is an illustration of a {\it shrinker principle} that uniqueness radiates out from a compact set. It implies that if one tangent flow at a future singular point is a cylinder, then all tangent flows are. We solve the gauge problem by solving a nonlinear system of PDEs. The PDE produces a diffeomorphism that fixes an appropriate gauge in the spirit of the slice theorem for group actions. We then show optimal bounds for the displacement function of the diffeomorphism. Strong rigidity relies on gauge fixing and several other new ideas. One of these is "propagation of almost splitting", another is quadratic rigidity in the right gauge, and a third is an optimal polynomial growth bound for PDEs that holds in great generality.Ricci flowRicci solitonsSingularitiesStability & uniquenessBoundaries & non-compact https://arxiv.org/abs/2109.04998arxiv:2109.04998Fri, 10 Sep 2021 00:00:00 GMTTobias Holck Colding, William P. Minicozzi — Analysis of non-compact manifolds almost always requires some controlled behavior at infinity. Without such, one neither can show, nor expect, strong properties. On the other hand, such assumptions restrict the possible applications and often too severely. In a wide range of areas non-compact spaces come with a Gaussian weight and a drift Laplacian. Eigenfunctions are L² in the weighted space allowing for extremely rapid growth. Rapid growth would be disastrous for many applications. Surprisingly, for very general tensors, manifolds and weights, we show the same polynomial growth bounds that Laplace and Hermite observed for functions on Euclidean space for the standard Gaussian. This covers all shrinkers for Ricci and mean curvature flows. These results open a door for understanding general non-compact spaces. It provides an analytic framework for doing nonlinear PDE on Gaussian spaces where previously the Gaussian weight allowed wild growth that made it impossible to approximate nonlinear by linear. It is key to bound the growth of diffeomorphisms of non-compact manifolds and is the key for solving the "gauge problem". The relative nature of the estimates and the slow growth in the bounds lead to "propagation of almost splitting" that is significantly stronger than pseudo locality and key for applications.Mean curvature flowRicci solitonsBoundaries & non-compactEigenvalues & spectrum