gpt 6 astra ultra attempt on Math-298

Huisken's Generic Singularity Conjecture for Mean Curvature Flow

Definitions and mathematical statement

Mean curvature flow evolves an embedded hypersurface by $\partial_tF=-H\nu$, with $H$ its scalar mean curvature. A singularity model is a tangent flow obtained by parabolic rescaling around a singular point. The conjectural permitted models are multiplicity-one round shrinking spheres and generalized cylinders, whose time-$-1$ slices are \[ S^k(\sqrt{2k})\times{\mathbb{R}}^{n-k}\subset{\mathbb{R}}^{n+1},\qquad1\le k\le n. \] The catalogue requests this for every singularity arising from an open dense generic set of smooth closed embedded initial hypersurfaces. A precise global statement requires the source's topology on initial hypersurfaces and its weak-flow continuation convention after singularities. Classifying stable self-shrinkers alone does not establish that all singularities of every member of an open dense set have these models.

Formulation and concept references: [S1].

Short English statement

For an open dense set of smooth closed initial hypersurfaces, must every mean-curvature-flow singularity look like a single shrinking round sphere or cylinder?

Research attempt

Scope, current progress and conventions.

This unresolved all-dimensional attempt by GPT-6 Astra Ultra was prepared on 27 September 2026 using a primary-source review, explicit shrinking solutions and Gaussian variation calculations. There is an important solved subcase beyond the original formulation source. Chodosh--Choi--Mantoulidis--Schulze [E1], Theorem 1.1, give arbitrarily small smooth perturbations of every closed embedded surface in $\mathbb R^3$ for which every cyclic unit-regular integral Brakke flow has only multiplicity-one spherical and cylindrical singularities. Their Remark 1(c) supplies openness as well as density. This resolves the surface case with their continuation convention. It is not an all-dimensional theorem. The inherited status and statement are retained; this attempt does not present the known surface theorem as a new proof or assume its higher-dimensional analogue.

For the direct computations take a smooth closed hypersurface before its first singular time, with outward normal on a round sphere and $H>0$ there. Thus the velocity is $-H\nu$. Assertions about tangent flows and continuation after singularities require weak-flow theory; the elementary computations below do not construct that theory.

Exact shrinking models and the normalization of their radii.

On a round $n$-sphere of radius $R(t)$, the principal curvatures are $1/R$, so \[ R'=-\frac nR,\qquad R(t)^2=R(0)^2-2nt. \] At its extinction time $T=R(0)^2/(2n)$, rescaling by $(T-t)^{-1/2}$ gives the sphere of radius $\sqrt{2n}$. For the product $S^k(R)\times\mathbb R^{n-k}$, exactly $k$ principal curvatures equal $1/R$ and the others vanish. Hence $R'=-k/R$, and the analogous rescaled radius is $\sqrt{2k}$. These computations reproduce the root statement with its factor $2$, rather than a convention-dependent radius $\sqrt{k}$.

A time-$-1$ shrinker obeys \[ H=\frac{\langle x,\nu\rangle}{2}. \tag{1} \] For the cylinder the right side is $R/2$, while the left is $k/R$; $1$ again gives $R^2=2k$. The flat case $k=0$ is a stationary plane, which models a regular multiplicity-one point rather than one of the genuine singularities requested in the statement. These explicit flows produce the permitted models. They do not prove that another initial hypersurface can only develop these models.

Deriving the Gaussian monotonicity identity with the chosen signs.

Fix $(x_0,t_0)$ and write $r=x-x_0$, $\tau=t_0-t>0$. Define \[ \rho(x,t)=(4\pi\tau)^{-n/2}\exp\left(-\frac{|r|^2}{4\tau}\right), \qquad \Phi(t)=\int_{M_t}\rho\,d\mu_t. \] The area element evolves by $\partial_t d\mu=-H^2d\mu$. Differentiating $\rho$ along the moving hypersurface gives \[ \frac{1}{\rho}\frac{d\rho}{dt} =\frac n{2\tau}-\frac{|r|^2}{4\tau^2} +\frac{H\langle r,\nu\rangle}{2\tau}. \] For the tangential vector $r^T$, direct differentiation yields $\operatorname{div}_M r^T=n-H\langle r,\nu\rangle$ and $\nabla_M\rho=-\rho r^T/(2\tau)$. Since $M_t$ is closed, \[ 0=\int_{M_t}\operatorname{div}_M(\rho r^T) =\int_{M_t}\rho\left(n-H\langle r,\nu\rangle -\frac{|r^T|^2}{2\tau}\right). \] Subtracting this zero identity with coefficient $1/(2\tau)$ from the differentiated integral completes the square: \[ \Phi'(t)=-\int_{M_t} \left(H-\frac{\langle r,\nu\rangle}{2\tau}\right)^2 \rho\,d\mu_t\leq0. \tag{2} \] This verifies the sign in $1$ independently of a remembered convention. For the centered shrinking sphere with $t_0=T$, the integrand vanishes identically and $\Phi$ is constant. A cylinder satisfies the same identity after the usual integrable Gaussian cutoff in its Euclidean directions.

Equation $2$ explains why a sufficiently regular rescaled limit with constant Gaussian density is a self-shrinker. Establishing existence, compactness, integrality and the appropriate equality case for arbitrary tangent flows uses additional theorems. Even after those inputs, $1$ is a differential equation admitting a larger class of solutions than the desired spheres and cylinders. Monotonicity is therefore a reduction to shrinkers, not their classification or generic avoidance.

Gaussian mass detects multiplicity as well as shape.

For a hypersurface $\Sigma$ put \[ F(\Sigma)=(4\pi)^{-n/2} \int_\Sigma e^{-|x|^2/4}\,d\mu. \] On $\Sigma_k=S^k(\sqrt{2k})\times\mathbb R^{n-k}$, the flat Gaussian integral cancels the factor from the Euclidean directions, leaving \[ F(\Sigma_k)=|S^k(1)| \left(\frac{k}{2\pi}\right)^{k/2}e^{-k/2}. \tag{3} \] In particular $F(\Sigma_1)=\sqrt{2\pi/e}$ and $F(\Sigma_2)=4/e$. A varifold consisting of the same support with integer multiplicity $m$ has Gaussian mass $mF(\Sigma_k)$. Thus recognizing the round support of a tangent object does not establish multiplicity one. A proof excluding other smooth shrinker shapes would still leave this separate obstruction. Small perturbations of initial data cannot simply be presumed to remove multiplicity without a theorem linking the perturbations to the weak tangent-flow measure.

A spherical second-variation calculation and a stability trap.

For a smooth shrinker, the normal second variation at fixed Gaussian center and scale has the form \[ F''(f\nu,f\nu)=-(4\pi)^{-n/2}\int_\Sigma fLf\,e^{-|x|^2/4}, \quad L=\Delta_\Sigma-\tfrac12\langle x^T,\nabla_\Sigma\rangle +|A|^2+\tfrac12. \tag{4} \] Here $4$ is the standard Gaussian variational formula used in [S1]; we compute its spectrum only on the round sphere. At radius $\sqrt{2n}$, $x^T=0$ and $|A|^2=1/2$, so $L=\Delta+1$. The spherical harmonics of degree $\ell$ have Laplace eigenvalue $\ell(\ell+n-1)/(2n)$, and hence $L$-eigenvalue \[ \mu_\ell=1-\frac{\ell(\ell+n-1)}{2n}. \tag{5} \] The constant mode has $\mu_0=1$, the linear modes have $\mu_1=1/2$, and every mode with $\ell\geq2$ has $\mu_\ell<0$. Thus the quadratic form in $4$ is positive on the subspace orthogonal to constants and linear harmonics, while it is negative in the dilation and translation normal directions. The scale test can also be seen directly: for a round radius-$R$ sphere, $\log F$ differs from $n\log R-R^2/4$ by a constant, whose second derivative at $R=\sqrt{2n}$ equals $-1$.

In particular the statement "the sphere minimizes the fixed-center Gaussian functional under every variation" is false. Generic singularity stability must account for the freedom to recenter and rescale. Entropy is the supremum of Gaussian functionals over those parameters, not the single fixed functional in $4$. The sphere calculation is consistent with the correct treatment of symmetry modes, but does not independently establish the global entropy classification in [S1]. No spherical spectral calculation classifies arbitrary noncompact shrinkers.

The missing step is global and involves quantifiers.

Suppose for each unwanted shrinker one finds a nearby initial perturbation that avoids that particular model near one spacetime point. This has the form "for each obstruction there exists a perturbation." The target requires one perturbed initial hypersurface whose entire continued flow avoids every unwanted model at every singular point. Passing between these quantifiers needs compactness, control of new singularities and a mechanism to globalize the local avoidance. The new flow might develop an unwanted singularity elsewhere, or a sequence of models might accumulate outside the region where a chosen local estimate is uniform.

A decreasing Gaussian quantity alone does not force termination of a perturbation procedure. Positive drops $2^{-j}$ have a finite sum, so infinitely many strict decreases can fit within a bounded initial budget. One needs a suitable uniform drop on controlled classes, a finite stratification, or another proved argument ruling out an infinite obstruction sequence. Such hypotheses must be supplied at the dimensions, entropy bounds and regularity levels under discussion; they cannot be imported from the surface case without checking them.

There is also a topology distinction. A countable intersection of open dense sets can be dense without being open, as the irrational numbers in $\mathbb R$ illustrate. Therefore a Baire-category construction of flows avoiding successively many bad events, even if valid, does not alone prove the root's open dense conclusion. Openness requires stability of the good global singularity behavior under perturbation of initial data. The source [E1] verifies the stronger conclusion in its surface setting; this attempt has no corresponding all-dimensional stability theorem or replacement for its dimension-specific inputs.

Verification and outcome.

The local checks verify the sphere and cylinder radius equations, the Gaussian constants in $3$, the exact rational eigenvalues in $5$, and the monotonicity square on evolving round spheres for arbitrary future Gaussian time. These tests would catch a missing factor of two, an incorrect curvature sign or the confusion between fixed Gaussian stability and adjustment of the symmetry parameters. They do not test weak-flow compactness or genericity.

Outcome: UNRESOLVED in the all-dimensional scope. The surface case is credited to existing work. The first missing input in the present approach is an all-dimensional global perturbation and stability theorem, with multiplicity control and a specified weak-flow continuation, that promotes local shrinker avoidance to the entire open dense conclusion. The displayed model and variation calculations establish only the stated partial checks and reductions.

Sources

• S1 T. H. Colding, W. P. Minicozzi II, 'Generic mean curvature flow I; generic singularities', arXiv:0908.3788 $2009$; Ann. of Math. $2$ 175 $2012$ 755-833. https://arxiv.org/abs/0908.3788. Formulation source.
• E1 Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis and Felix Schulze, Revisiting generic mean curvature flow in $\mathbb R^3$, arXiv:2409.01463; Journal für die reine und angewandte Mathematik 835 $2026$, 219--232. https://arxiv.org/html/2409.01463. Primary text, Theorem 1.1 and Remark 1(c), consulted 27 September 2026 for the resolved surface case, its open dense conclusion and its specified Brakke-flow convention.