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Bamler--Kleiner recently proved a multiplicity-one theorem for mean curvature flow in $\mathbb{R}^3$ and combined it with the authors' work on generic mean curvature flows to fully resolve Huisken's genericity conjecture. In this paper we show that a short density-drop theorem plus the Bamler--Kleiner multiplicity-one theorem for tangent flows at the first nongeneric singular time suffice to resolve Huisken's conjecture -- without relying on the strict genus drop theorem for one-sided ancient flows previously established by the authors., Comment: Comments welcome!

In this paper, we prove that an ancient smooth curve shortening flow with finite-entropy embedded in $\mathbb{R}^2$ has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplity $m\geq 3$ exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.

We construct an $I$-family of ancient graphical mean curvature flows over a minimal hypersurface in $\mathbb{R}^{n+1}$ of finite total curvature with the Morse index $I$ by establishing exponentially fast convergence in terms of $|x|^2-t$. As a corollary, we show that these ancient flows have finite total curvature and finite mass drop. Moreover, one family of these flows is mean convex by a pointwise estimate., Comment: All comments are welcome

We study the regularity and the growth rates of solutions to two-dimensional Monge-Amp\`ere equations with the right-hand side exhibiting polynomial growth. Utilizing this analysis, we demonstrate that the translators for the flow by sub-affine-critical powers of the Gauss curvature are smooth, strictly convex entire graphs. These graphs exhibit specific growth rates that depend solely on the power of the flow., Comment: A part of this work was introduced in arXiv:2104.13186v1; however, we have separated and further elaborated the result to accommodate issues that will be dealt in the revision of arXiv:2104.13186. In v2, we generalized the main theorems slightly so that they apply for broader class of equations

We classify the smooth linearly stable self-similar solutions of the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ under an integral condition for all $p>1$. As a corollary, we prove that finite time blowing up solutions of this equation on a bounded convex domain with $u(\cdot,0)\geq 0$ and $u_t(\cdot,0)\geq 0$ converges to a constant after rescaling at the blow-up point for all $p>1$., Comment: All comments are welcome

This paper shows the existence of convex translating surfaces under the flow by the $\alpha$-th power of Gauss curvature for the sub-affine-critical regime $ 0 < \alpha < 1/4$. The key aspect of our study is that our ansatz at infinity is the graph of homogeneous functions whose level sets are closed curves shrinking under the flow by the $\frac{\alpha}{1-\alpha}$-th power of curvature. For each ansatz, we construct a family of translating surfaces generated by the Jacobi fields with effective growth rates. Moreover, the construction shows quantitative estimate on the rate of convergence between different translators to each other, which is required to show the continuity of the family. As a result, the family is regarded as a topological manifold. The construction in this paper will become the ground of forthcoming research, where we aim to prove that every translating surface must correspond to one of the solutions obtained herein, classifying translating surfaces and identifying the topology of the moduli space., Comment: A part of this work was introduced in arXiv:2104.13186v1; however, we have separated and further elaborated on this existence theory. See footnote 3 on page 3 of the manuscript. In v2, we added Corollary A.1

We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp $C^0$-estimate and a quadratic concavity estimate for the profile function of two-convex ancient ovals, which are crucial in their papers as well as a slew of subsequent papers on ancient solutions of mean curvature flow and Ricci flow. In this paper, we derive a sharp gradient estimate, which enhances their $C^0$-estimate, and a sharp Hessian estimate, which can be viewed as converse of their quadratic concavity estimate. Motivated by our forthcoming work on ancient noncollapsed flows in $\mathbb{R}^4$, we derive these estimates in the context of ancient ovals in $\mathbb{R}^3$ and noncollapsed translators in $\mathbb{R}^4$, though our methods seem to apply in other settings as well., Comment: 18 pages

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure $\mu$ with a positive smooth density $f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p \le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $\mathbb{R}^3$., Comment: 25 pages, to appear in Adv. Math

We show that the mean curvature flow of a generic closed surface in $\mathbb{R}^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot arise generically., Comment: Comments welcome!

In this paper, we classify all noncollapsed singularity models for the mean curvature flow of 3-dimensional hypersurfaces in $\mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $\mathbb{R}^4$ is either $\mathbb{R}\times$2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White., Comment: 98 pages

Some of the most worrisome potential singularity models for the mean curvature flow of $3$-dimensional hypersurfaces in $\mathbb{R}^4$ are noncollapsed wing-like flows, i.e. noncollapsed flows that are asymptotic to a wedge. In this paper, we rule out this potential scenario, not just among self-similarly translating singularity models, but in fact among all ancient noncollapsed flows in $\mathbb{R}^4$. Specifically, we prove that for any ancient noncollapsed mean curvature flow $M_t=\partial K_t$ in $\mathbb{R}^4$ the blowdown $\lim_{\lambda\to 0} \lambda\cdot {K_{t_0}}$ is always a point, halfline, line, halfplane, plane or hyperplane, but never a wedge. In our proof we introduce a fine bubble-sheet analysis, which generalizes the fine neck analysis that has played a major role in many recent papers. Our result is also a key first step towards the classification of ancient noncollapsed flows in $\mathbb{R}^4$, which we will address in a series of subsequent papers., Comment: 40 pages

We classify the surfaces translating under the flows by sub-affine-critical powers of the Gauss curvature. This, in particular, lists all translating solitons possibly model Type II singularities for convex closed solutions in all positive powers. The surfaces are entire graphs, and therefore our result corresponds to the Liouville theorem for the degenerate Monge--Amp\`ere equations $\det D^2 u=(1+|Du|^2)^{2-\frac{1}{2\alpha}}$ on $\mathbb{R}^2$ in the range $0<\alpha <1/4$. The result also reveals that the moduli spaces of solutions are homeomorphic to either Euclidean spaces or cylinders., Comment: In this revision, we (i) fixed a gap by proving a rigidity result in Section 3.5, and (ii) improved the result by determining the topology of the moduli space of translators in Section 5

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in $\mathbb{R}^3$ with entropy $\leq 2$ and to all closed hypersurfaces in $\mathbb{R}^4$ with entropy $\leq \lambda(\mathbb{S}^1 \times \mathbb{R}^2)$. When combined with recent work of Daniels-Holgate, this strengthens Bernstein-Wang's low-entropy Schoenflies-type theorem by relaxing the entropy bound to $\lambda(\mathbb{S}^1 \times \mathbb{R}^2)$. Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt-Simon and Smale)., Comment: 18 pages. Final version, to appear in Duke Mathematical Journal

We classify closed convex $\alpha$-curve shortening flows for sub-affine-critical powers $\alpha \leq \frac{1}{3}$. In addition, we show that closed convex smooth finite entropy $\alpha$-curve shortening flows with $\frac{1}{3}<\alpha$ is a shrinking circle. After normalization, the ancient flows satisfying the above conditions converge exponentially fast to smooth closed convex shrinkers at the backward infinity. In particular, when $\alpha=\frac{1}{k^2-1}$ with $3\leq k \in \mathbb{N}$, the round circle shrinker has non-trivial Jacobi fields, but the ancient flows do not evolve along the Jacobi fields.

We show the existence of non-homothetic ancient flows by powers of curvature embedded in $\mathbb{R}^2$ whose entropy is finite. We determine the Morse indices and kernels of the linearized operator of shrinkers to the flows and construct ancient flows by using unstable eigenfunctions of the linearized operator., Comment: Minor changes in some paragraphs. We add several new references in this version

We address the classification of ancient solutions to the Gauss curvature flow under the assumption that the solutions are contained in a cylinder of bounded cross section. For each cylinder of convex bounded cross-section, we show that there are only two ancient solutions which are asymptotic to this cylinder: the non-compact translating soliton and the compact oval solution obtained by gluing two translating solitons approaching each other from time $-\infty$ from two opposite ends., Comment: v2 Mistypes corrected. Lemma 2.6 has been added. The proof of Proposition 3.8 has been simplified v3 Section 6 is revised. Mistypes were corrected

We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in $\mathbb{R}^{4}$ is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons., Comment: Final version, to appear in Invent. Math

In this paper, we prove the mean-convex neighborhood conjecture for neck singularities of the mean curvature flow in $\mathbb{R}^{n+1}$ for all $n\geq 3$: we show that if a mean curvature flow $\{M_t\}$ in $\mathbb{R}^{n+1}$ has an $S^{n-1}\times \mathbb{R}$ singularity at $(x_0,t_0)$, then there exists an $\varepsilon=\varepsilon(x_0,t_0)>0$ such that $M_t\cap B(x_0,\varepsilon)$ is mean-convex for all $t\in(t_0-\varepsilon^2,t_0+\varepsilon^2)$. As in the case $n=2$, which was resolved by the first three authors in arXiv:1810.08467, the existence of such a mean-convex neighborhood follows from classifying a certain class of ancient Brakke flows that arise as potential blowup limits near a neck singularity. Specifically, we prove that any ancient unit-regular integral Brakke flow with a cylindrical blowdown must be either a round shrinking cylinder, a translating bowl soliton, or an ancient oval. In particular, combined with a prior result of the last two authors, we obtain uniqueness of mean curvature flow through neck singularities. The main difficulty in addressing the higher dimensional case is in promoting the spectral analysis on the cylinder to global geometric properties of the solution. Most crucially, due to the potential wide variety of self-shrinking flows with entropy lower than the cylinder when $n\geq 3$, smoothness does not follow from the spectral analysis by soft arguments. This precludes the use of the classical moving plane method to derive symmetry. To overcome this, we introduce a novel variant of the moving plane method, which we call "moving plane method without assuming smoothness" - where smoothness and symmetry are established in tandem., Comment: 83 pages, to appear in Inventiones

We study closed ancient solutions to gradient flows of elliptic functionals in Riemannian manifolds, including mean curvature flow and harmonic map heat flow. Our work has various consequences. In all dimensions and codimensions, we classify ancient mean curvature flows in S^n with low area: they are steady or shrinking equatorial spheres. In the mean curvature flow case in S^3, we classify ancient flows with more relaxed area bounds: they are steady or shrinking equators or Clifford tori. In the embedded curve shortening case in S^2, we completely classify ancient flows of bounded length: they are steady or shrinking circles., Comment: Final version; to appear in Amer. J. Math

We address the asymptotic behavior of the $\alpha$-Gauss curvature flow, for $\alpha >1/2$, with initial data a complete non-compact convex hypersurface which is contained in a cylinder of bounded cross section. We show that the flow converges, as $t \to +\infty$, locally smoothly to a translating soliton which is uniquely determined by the asymptotic cylinder of the initial hypersurface., Comment: To be published in Adv. Math

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a positive $\varepsilon=\varepsilon(X)>0$ such that the flow is mean convex in a space-time neighborhood of size $\varepsilon$ around $X$. The major difficulty is to promote the infinitesimal information about the singularity to a conclusion of macroscopic size. In fact, we prove a more general classification result for all ancient low entropy flows that arise as potential limit flows near $X$. Namely, we prove that any ancient, unit-regular, cyclic, integral Brakke flow in $\mathbb{R}^3$ with entropy at most $\sqrt{2\pi/e}+\delta$ is either a flat plane, a round shrinking sphere, a round shrinking cylinder, a translating bowl soliton, or an ancient oval. As an application, we prove the uniqueness conjecture for mean curvature flow through spherical or cylindrical singularities. In particular, assuming Ilmanen's multiplicity one conjecture, we conclude that for embedded two-spheres the mean curvature flow through singularities is well-posed., Comment: 74 pages, to appear in Acta

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $\mathbb{R}^2$ under the $\alpha$-curve shortening flow for exponents $\alpha >\frac12$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $\alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $\alpha=1$, and we prove for all exponents up to the critical case $\alpha>\frac12$., Comment: 28 pages, 0 figure

We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha \geq \frac{1}{n+2}$. For $\alpha > \frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha=\frac{1}{n+2}$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling., Comment: final version, to appear in Acta Math

We derive local $C^{2}$ estimates for complete non-compact translating solitons of the Gauss curvature flow in $\mathbb{R}^3$ which are graphs over a convex domain $\Omega$. This is closely is related to deriving local $C^{1,1}$ estimates for the degenerate Monge-Amp\'ere equation. As a result, given a weakly convex bounded domain $\Omega$, we establish the existence of a $C^{1,1}_{\text{loc}}$ translating soliton. In particular, when the boundary $\partial \Omega$ has a line segment, we show the existence of flat sides of the translator from a local a'priori non-degeneracy estimate near the free-boundary.

We show the uniqueness of strictly convex closed smooth self-similar solutions to the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$. We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the $\alpha$-Gauss curvature flow with $(1/n) < \alpha < 1+(1/n)$ shrinks a strictly convex closed smooth hypersurface to a round sphere.

We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature for all time. In particular, the initial graph is not necessarily differentiable.

We consider the $Q_k$ flow on complete non-compact graphs. We prove that a complete graph evolves by the $Q_k$ curvature up to some time $T$ depending on the radius of a sphere enclosed by the initial graph.