More Like this

1. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with type-II curvature blowup

   https://doi.org/10.1088/1361-6544/ac15a9  

   Garfinkle, David; Isenberg, James; Knopf, Dan; Wu, Haotian (August 2021, Nonlinearity)

   Abstract We present a numerical study of the local stability of mean curvature flow (MCF) of rotationally symmetric, complete noncompact hypersurfaces with type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs ‘numerically global’ (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviours under MCF. Firstly, there is a ‘near’ class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a type-II rate no slower than ( T − t ) −1 . Secondly, there is a ‘far’ class of initial data which lead to solutions developing a local type-I nondegenerate neckpinch under MCF. These numerical findings further suggest the existence of a ‘critical’ class of initial data which conjecturally lead to MCF of noncompact hypersurfaces forming local type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than ( T − t ) −1 . 

   more » « less
2. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

   more » « less
3. Mean curvature flow from conical singularities

   https://doi.org/10.1007/s00222-024-01296-8  

   Chodosh, Otis; Daniels-Holgate, J M; Schulze, Felix (December 2024, Inventiones mathematicae)

   Abstract We prove Ilmanen’s resolution of point singularities conjecture by establishing short-time smoothness of the level set flow of a smooth hypersurface with isolated conical singularities. This shows how the mean curvature flow evolves through asymptotically conical singularities. Precisely, we prove that the level set flow of a smooth hypersurface$$M^{n}\subset \mathbb{R}^{n+1}$$ $M^n\subset R^{n+1}$,$$2\leq n\leq 6$$ $2\le n\le 6$, with an isolated conical singularity is modeled on the level set flow of the cone. In particular, the flow fattens (instantaneously) if and only if the level set flow of the cone fattens. 

   more » « less
4. Mean curvature flow

   https://doi.org/10.1090/S0273-0979-2015-01468-0  

   Colding, Tobias; Minicozzi, William; Pedersen, Erik (January 2015, Bulletin of the American Mathematical Society)

   Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur under the flow, what the flow looks like near these singularities, and what this implies for the structure of the singular set. At the end, we will briefly discuss how one may be able to use the flow in low-dimensional topology. 

   more » « less
5. Existence of hypersurfaces with prescribed mean curvature I – generic min-max

   Zhou, Xin; Zhu, Jonathan J. (April 2020, Cambridge journal of mathematics)

   We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersurface of multiplicity one. More precisely, we show that our previous min-max theory, developed for constant mean curvature hypersurfaces, can be extended to construct min-max prescribed mean curvature hypersurfaces for certain classes of prescription function, including a generic set of smooth functions, and all nonzero analytic functions. In particular we do not need to assume that h has a sign. 

   more » « less