Unique tangent flows: Gaussian area, cylindrical openings, and finite rescaled length

Source and verification

This article explains family 355 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

At a singularity of mean-curvature flow, magnifying space and time can reveal a simpler limiting motion. Compactness often gives such a limit along a sequence of magnifications. It does not say that a different sequence finds the same surface, or even the same axis of a cylindrical surface.

This family proves uniqueness at the first singular time of a smooth closed embedded surface in three-dimensional Euclidean space. Its conclusion keeps the singular point and ambient coordinates fixed. The proof combines Gaussian gradient flow, an obstruction to changing cylindrical end types, and a propagation argument that controls every sufficiently large rescaling. (OpenAI, 2026)

The prerequisites are differential calculus on surfaces, ordinary differential equations, basic elliptic and parabolic PDE, and power series. We introduce the geometric normalization first. The intricate stationary construction is presented through its principal estimates and the mechanisms linking them; this is a selected-section source audit, not an independent verification of the full proof.

🏷️ Mean-curvature flow and parabolic magnification

Let $M_0\subset {\mathbb{R}}^3$ be a smooth compact connected embedded surface without boundary. Under mean-curvature flow, the normal velocity is the mean curvature vector $\mathbf{H}$, the trace of the second fundamental form. Let $M_t$, $0\le t<T$, be its maximal smooth flow. The first singular time is the endpoint $T$ at which smooth continuation fails.

Fix a singular point $(x_0,T)$. The natural magnifications are

$$\begin{array}{c}{M}_s^{\lambda }=\lambda (M_{T+{\lambda }^{-2}s}-x_0),\\ s<0.\end{array}$$

The time factor ${\lambda }^{-2}$ matches the fact that curvature has inverse-length units. A tangent flow is a limit extracted as $\lambda \to \infty$. Its multiplicity records how many sheets contribute to the limiting area measure. Smoothness of the limiting support alone does not guarantee multiplicity one or smooth convergence of the approximating surfaces.

A self-shrinker is a surface $S$ satisfying

$$\mathbf{H}+\frac{1}{2}{X}^{\perp }=0,$$

where $X^{\perp }$ is the normal projection of the position vector. It generates the homothetic flow $\sqrt{-s}S$. Familiar time-minus-one examples are the sphere of radius $2$, the round cylinder of radius $\sqrt{2}$, and a plane through the origin. These examples do not exhaust the tangent models considered in the manuscript.

Uniqueness in the original ambient coordinates

For every singular point at the first singular time of the flow above, the manuscript proves that there is a smooth properly embedded self-shrinker $S$ such that the full family $M_s^{\lambda }$ converges locally smoothly in space, with multiplicity one, to $\sqrt{-s}S$. The convergence is uniform for $s$ in compact subintervals of $(-\infty ,0)$ and uses the original ambient coordinates, including the axes and position of the limit. (OpenAI, 2026)

Properness means that the intersection of $S$ with a compact ambient set is compact. There is no mean-convexity assumption and no prescribed tangent model. The conclusion concerns a fixed center, the smooth evolution before the first singular time, and negative rescaled times. It does not assert a unique continuation through the singularity or uniqueness for moving-center blowups.

The paper also identifies every backward integral Brakke tangent at every negative time, under its explicitly stated all-endpoint convention. We explain that additional measure-theoretic conclusion at the end.

🏷️ Gaussian area and the finite-length objective

Translate the singular spacetime point to $(0,0)$ and write the original times as $t<0$. Introduce logarithmic rescaled time:

$$M(\tau )=e^{\tau /2}{M}_{-e^{-\tau }}.$$

Self-shrinkers are stationary for this rescaled flow. Its normal velocity is the shrinker defect $\Phi =\mathbf{H}+X^{\perp }/2$. The relevant energy is Gaussian area, with an inessential constant normalization omitted:

$$\begin{array}{rl}F(M)& ={\int }_M{e}^{-|X{|}^2/4}d\mu ,\\ d(\tau {)}^2& ={\int }_{M(\tau )}|\Phi {|}^2{e}^{-|X{|}^2/4}d\mu .\end{array}$$

Huisken monotonicity gives $-F^{\prime }=d^2$. Fix one extracted tangent section $S$. Uniform quadratic area bounds control the Gaussian tails, so compact convergence along its extracting sequence identifies the full energy limit. Thus

$$\begin{array}{c}\mathcal{E}(\tau )=F(M(\tau ))-F(S)\searrow 0,\\ {\mathcal{E}}^{\prime }=-d^2.\end{array}$$

Finite energy dissipation gives $\int d^2<\infty$. Convergence requires the stronger finite-length estimate $\int d<\infty$. The difference is substantial: $d(\tau )=1/(1+\tau )$ has integrable square but divergent integral.

The model mechanism is a gradient inequality

$${\mathcal{E}}^{1-\theta }\le Cd,0<\theta <1/2.$$

Where $\mathcal{E}>0$, it gives

$$-\frac{d}{d\tau }{\mathcal{E}}^{\theta }=\theta {\mathcal{E}}^{\theta -1}{d}^2\ge \frac{\theta }{C}d.$$

Integration then bounds the remaining length by a power of the remaining energy. The actual proof must establish a version with spatial truncation errors and ensure that its graphical hypotheses persist. Those two tasks account for much of the argument.

The available tangent geometry

The manuscript imports multiplicity-one and tangent-structure results of Bamler–Kleiner, together with local smooth regularity for multiplicity-one limits. They supply a properly embedded tangent shrinker with bounded geometry, finite genus, and finitely many separated ends, each asymptotic to a regular cone or a round cylindrical tail. The cited end theorem supplies the finite end decomposition directly; finite genus by itself would not justify assuming finite topology. (OpenAI, 2026)

A shrinker with a cylindrical end need not already have been identified as an exact cylinder. The stationary argument treats all conical and cylindrical ends simultaneously and does not assume such a cylinder-rigidity theorem.

🏷️ Compact observations and cylindrical openings

For a normal graph over $S$, the linearized shrinker operator is

$$L={\Delta }_S-\frac{1}{2}{X}^{\top }\cdot {\nabla }_S+\frac{1}{2}+|\mathrm{II}{|}^2.$$

It is self-adjoint in Gaussian $L^2$ and has a finite-dimensional kernel. The source proves this using Gaussian moment estimates and conjugation to a Schrödinger operator whose potential grows quadratically at infinity. This confines the relevant low-frequency behavior despite the noncompact ends.

To handle the kernel, choose finitely many compact observations

$$\begin{array}{c}{P}_{i}u={\int }_{S}u{\chi }_i{e}^{-|X{|}^2/4}d\mu ,\\ {\chi }_i\in C_c^{\infty }(S).\end{array}$$

The observation list is chosen so that the augmented map $(u,\lambda )\mapsto (Lu+P^{\ast }\lambda ,Pu)$ is invertible. Prescribing observations now selects normalized stationary solutions, with compactly supported source terms represented by the multipliers $\lambda$. A member of this family is an exact shrinker outside the observation core; it need not be an exact shrinker on the core unless its multipliers vanish.

The parameters split into $(b,x)$. The $b$ parameters preserve end types and include end tilts and variations of conical links. There is an analytic family $S_b$ at $x=0$. There is one opening coordinate in $x$ for each cylindrical end. Its leading normal displacement has quadratic axial growth, with coefficient ${\beta }_j$, where

$$\beta (b,x)=M(b)x$$

and $M(b)$ is invertible. The source constructs successive polynomial-growth Taylor coefficients in $x$, but does not assert convergence of their infinite series.

The circular comparison explains the problem:

$$R_c(z{)}^2=2(1+\beta z^2).$$

For $\beta <0$, the comparison radius vanishes at $Z_0=(-\beta {)}^{-1/2}$. This happens farther and farther away as the opening tends to zero. The Gaussian weight at that location is

$$e^{-Z_0^2/4}=e^{-1/(4|\beta |)}.$$

This is smaller than every fixed power of $|\beta |$. Analytic dependence for decaying graphs therefore does not automatically settle what happens in opening directions that change the end geometry at that remote scale.

Coefficientwise Gaussian integration gives a formal action $\mathcal{F}(b,x)$: a power series in $x$ whose coefficients are analytic in $b$. A formal critical arc is a power-series path $(b(s),x(s))$ through zero for which every coefficient of $\nabla \mathcal{F}(b(s),x(s))$ vanishes. The central stationary assertion is that every such arc has $x(s)\equiv 0$. It excludes critical opening deformations, while allowing the end-preserving parameters to remain. (OpenAI, 2026)

🏷️ An exponentially small obstruction

Suppose a critical arc has first nonzero opening term $x(s)=s^{k}v+O(s^{k+1})$, with $v\ne 0$. The proof constructs two stationary actions $G_{+}$ and $G_{-}$ using paths passing on opposite sides of a collapsing cylindrical end. The equations have complex coefficients on real spatial parameter domains. An arbitrary smooth spatial graph is not being analytically continued to complex spatial points.

The two actions have the same formal expansion, with factorial-type control of coefficients and remainders. Such bounds are often called Gevrey estimates. For an opening of order $k$, their coefficient growth has the form $C{A}^n\Gamma (1+n/k)$. The source obtains this by telescoping stopped problems: a problem stopped at radius $R$ has a parameter disk of radius comparable to $R^{-2/k}$, while consecutive stopped actions differ by $O(e^{-c{R}^2})$. Cauchy estimates and summation over radii yield the factorial bounds.

Finite-dimensional implicit solves select nearly critical parameters for each action. These parameters need not coincide. Transporting both actions to their common midpoint is essential: their gradients are already exponentially small, so Taylor transport has a quadratic parameter-error cost. After a small fixed change of complex angle, the source obtains

$$|G_{+}-G_{-}|=O(e^{-1.2A_{\mathrm{lead}}}),$$

where $A_{\mathrm{lead}}$ is the smallest leading Gaussian action scale of the collapsing ends. This is the upper estimate that a hypothetical critical arc would force.

A translator equation and its nonzero lateral difference

Near collapse, rescaling the height equation produces the radial translating-soliton ODE

$$l^{\prime }=(1+l^2)(1-l/y).$$

Here $l$ is the radial slope. The relevant exterior solutions satisfy $l/y\to 1$. Put $\tau =y^2/2$ and $q=1-l/y$. Then

$$q_{\tau }+q=2q^2-q^3+\frac{1-2q}{2\tau }.$$

The two lateral solutions have a common formal expansion $q={\sum }_{n\ge 1}{b}_n{y}^{-2n}$, with positive coefficients obeying

$$2^{n-1}(n-1)!\le b_n\le 3^{n-1}(n-1)!.$$

The series is divergent. Divergence alone would not establish the needed nonzero difference. The manuscript additionally proves factorial remainder estimates and argues that agreement of the two lateral solutions would analytically continue their positive-coefficient Borel series through its positive convergence radius. The positive-coefficient singularity principle rules that out.

Subtracting the two exact slope equations then gives a scalar linear equation whose coefficient is $-y+3/y+O(y^{-3})$. Integration yields

$$\begin{array}{c}{l}_{+}(y)-l_{-}(y)\\ =c_0{y}^3{e}^{-y^2/2}(1+O(y^{-2})),\\ c_0\ne 0.\end{array}$$

This nonzero difference is converted into an action difference by boundary momentum. The exact finite-scale first variation is used before taking the translator limit, so large boundary contributions cancel in a controlled expression. The resulting end contribution has scale $Z_{\ast }^{-2}{e}^{-Z_{\ast }^2/4}$, where $Z_{\ast }$ is a corrected collapse height. (OpenAI, 2026)

Why different ends cannot cancel

The action difference is a sum over collapsing ends. Its coefficient has the same nonzero sign and phase for the consistently ordered pair of bypasses on every end. But that fact alone would not control complex exponential factors with different phases.

The source therefore computes the corrected phase through constant order. In particular,

$$\begin{array}{rl}{Z}_{\ast }^2=& -\frac{1}{\beta }-8\log Z_0\\ & +\frac{c_2(b,x)}{{\beta }^2}\\ & +2+2\log 2+o(1).\end{array}$$

Here $c_2$ is a coefficient obtained from the second mean jet, rather than an unspecified error. Distinct negative-power parts of $-1/{\beta }_j(s)$ separate in real part after a generic small change of evaluation angle. Among ends with the same dominant negative-power part, the corrected phase differences tend to real constants. Their relative exponential weights consequently tend to strictly positive numbers. Thus the dominant terms do not cancel.

The resulting lower bound is $|G_{+}-G_{-}|\ge e^{-1.10A_{\mathrm{lead}}}$ for sufficiently small $s$, contradicting the upper bound with exponent $1.2$.

Even-order arcs require care. If all leading opening coefficients are positive, replacing a real parameter $s$ by $-s$ does not reverse an even power. The proof instead evaluates at angle $\arg s=\pi /k$ on one continuously chosen sheet, and keeps both actions on that sheet. It does not assume that their actual parameters are real or that the actions are complex conjugates. The contradiction excludes every nonzero formal opening arc.

🏷️ A fixed Taylor polynomial gives the gradient estimate

The source next invokes a finite-dimensional transfer theorem, using a formal real arc criterion and analytic approximation. We state the resulting estimate without developing its algebraic machinery. Let $f_N$ be the truncation of $\mathcal{F}$ through opening degree $N$. There is one integer $m\ge 1$ such that, for every sufficiently large fixed $N$,

$$|x{|}^m\le C_N(|\nabla f_N(b,x)|+|x{|}^{N-1}).$$

The exponent $m$ is fixed before choosing $N$. One then chooses a single $N$ with $N-1>4\max \{m,1\}$ and applies the ordinary analytic Łojasiewicz gradient inequality to that one analytic function $f_N$. The proof never needs the infinite opening series to converge.

To compare this finite object with a genuine flow slice, cut off its polynomial graph at a radius $|x{|}^{-{\sigma }_N}$. Choosing ${\sigma }_N$ small makes the height and its first two spatial derivatives uniformly small; Gaussian decay makes the cutoff error smaller than the retained Taylor remainder. A shifted Gaussian inverse compares this graph with the actual surface, retaining the compact multipliers and the spatial cutoff.

The comparison only requires small height and slope for the actual graph. The quasilinear subtraction keeps its principal coefficients on the second derivatives of the difference; the remaining second derivatives belong to the small Taylor comparison graph. Thus an unproved small Hessian bound for the actual slice is not inserted into the estimate.

A complete graph to radius $R$ means that the entire surface intersection there is the specified single graph, with no extra sheet or component. For such a graph with sufficiently small height and slope over $S_b$, set

$$\mathcal{D}=d+e^{-(1-\eta )R^2/8}.$$

The localized result is

$$\begin{array}{c}|F(M)-F(S){|}^{1-\theta }\le C_{\eta }\mathcal{D},\\ |x|\le C_{\eta }{\mathcal{D}}^{\gamma },\\ \theta \in (0,1/2),\gamma >0.\end{array}$$

The exponents $\theta ,\gamma$ are independent of the arbitrarily small localization loss $\eta$. The denominator $8$ is the Gaussian norm scale; using the energy-tail denominator $4$ here would strengthen the estimate incorrectly. Weighted interpolation with a weaker, still integrable Gaussian controls action differences without assuming finite unweighted area of the noncompact shrinker.

If there are no cylindrical ends, there are no opening variables. The finite-power step is omitted, but the actual spatial cutoff and compact-source comparison remain necessary. (OpenAI, 2026)

🏷️ Propagation and radii chosen from past energy drops

A gradient inequality on one graph is not enough. The proof must preserve a complete graph as time advances and allow its radius to grow.

At a time when $d$ is very small relative to the available radius, elliptic estimates improve the graph on a slightly smaller ball. Graphical pseudolocality then propagates the entire graph for a fixed time, and positive-time smoothing supplies derivative bounds. The order is important: improve the old graph, propagate and smooth it, change the reference parameter $b$, then apply the next elliptic improvement. Curvature estimates at zero elapsed time are not assumed.

Shift the rescaled clock to start far along the original tangent sequence. For unit time intervals define

$$\begin{array}{rl}{a}_j& =(\mathcal{E}(j)-\mathcal{E}(j+1){)}^{1/2},\\ r_j& =\sqrt{8\log (1/a_j)}.\end{array}$$

Set $r_j=\infty$ if $a_j=0$. For a fixed $Q>1$ compatible with propagation, choose

$$R_j=\min \left\{Q^j{R}_{\mathrm{in}},\underset{0\le i\le j}{\min }{Q}^{j-i}{r}_i\right\}.$$

Then $R_{j+1}=\min \{Q{R}_j,r_{j+1}\}$. The new radius never demands more than propagation from the preceding radius supplies. Averaging selects ${\tau }_j\in [j+.2,j+.4]$ with $d({\tau }_j)\le \sqrt{5}{a}_j$, and the local inequality gives

$$\begin{array}{c}\mathcal{E}(j+1{)}^{1-\theta }\\ \le C\left(a_j+e^{-(1-\eta )R_j^2/8}\right).\end{array}$$

The observations satisfy $|(b,x{)}^{\prime }|\le Cd$. Changing the base surface over radius $R$ costs at most $C(1+R)|\Delta b|$. Both adjacent energy cutoffs in the radius recurrence are used to make this change small: quantities such as $a_j\sqrt{\log (1/a_j)}$ tend to zero.

Summing the localization errors

Choose $0<\eta <c<\theta$. Call an index good when its Gaussian tail is at most $a_j^{1-c}$. At good indices, concavity of a suitable power of $\mathcal{E}$ bounds $a_j$ by a telescoping energy decrement.

At bad indices put $\kappa =(1-\eta )/(1-c)>1$ and $q=Q^2>1$. The radius formula gives

$$a_j\le e^{-(\kappa /8)q^j{R}_{\mathrm{in}}^2}+\sum _{i<j}{a}_i^{\kappa q^{j-i}}.$$

Its own term is excluded because $a_j^{\kappa }<a_j$ when $0<a_j<1$. The remaining powers of earlier drops are summable and, when the initial energy is small, contribute less than half of the total sum. Absorption therefore gives a finite-prefix bound uniform in the number of intervals, and eventually

$$\sum _{j\ge 0}{a}_j<\infty ,{\int }_0^{\infty }d(\tau )d\tau <\infty .$$

This finite-prefix form also closes the graphical bootstrap without using future energy drops. If the core graph first failed at a finite time, propagation from the last available improved slice would cover the remaining time gap. Finite accumulated motion and interior interpolation keep the core strictly inside its tolerance, contradicting that first failure. Complete graphical coverage excludes an extra sheet entering the core.

Since $a_j\to 0$, the delayed radii tend to infinity. The core graph is Cauchy, and its limit is the originally chosen $S$ because the original extracting sequence still converges to $S$. Propagation between selected times and smoothing from a preceding slice give smooth convergence on every fixed compact set at all sufficiently large rescaled times. The compact observations remain in the original frame, so this controls rotational drift as well as shape. (OpenAI, 2026)

🏷️ Every scale and every negative-time measure

Restoring the original rescaled clock gives the exact identity

$$M_s^{\lambda }=\sqrt{-s}M(2\log \lambda -\log (-s)).$$

For $s$ in a compact negative-time interval, the argument on the right tends uniformly to infinity. The full smooth blowup conclusion follows, with fixed center and no additional rotations.

A weak spacetime measure limit initially identifies slice measures only for almost every time. To remove exceptional times, the manuscript specifies a Brakke inequality for every time pair $a<b$, with compact nonnegative tests allowed to be nonzero at both endpoints. It also requires local spacetime integrability of mass and squared generalized mean curvature. These are part of the tangent convention, not consequences of almost-everywhere equality alone.

Let ${\nu }_s$ be the smooth shrinking reference measure and ${\mu }_s$ a tangent representative agreeing with it almost everywhere. For a fixed nonnegative compact spatial test $\varphi$, subtract the reference transport equality from the all-endpoint Brakke inequality. It yields

$${\mu }_b(\varphi )-{\nu }_b(\varphi )\le {\mu }_a(\varphi )-{\nu }_a(\varphi ).$$

The difference is a finite nonincreasing function of time and is zero almost everywhere. At any negative time, sandwich it between earlier and later times where it is zero. It must therefore be zero there too. Compact tests determine Radon measures, proving equality at every negative time. The argument does not require finite global unweighted area.

🏷️ Key steps and proof outlines

The exponential obstruction must dominate all approximation errors

The stationary analysis compares two bypasses of each collapsing cylindrical end. Their formal expansions agree, so a finite Taylor calculation alone sees no difference. The translator analysis supplies a nonzero exponentially small difference, and boundary momentum transfers it to the stationary action.

To compare actions chosen at slightly different almost-critical parameters, transport both to a common midpoint. Their gradients are already exponentially small, making this transport error quadratic in the parameter discrepancy. The resulting upper estimate decays like $e^{-1.2A}$, while the matched end calculation gives a lower bound $e^{-1.10A}$. The separation of exponents supplies the contradiction. The second mean jet and constant phase terms are retained so that ends with the same leading exponential have positive relative weights and cannot cancel. (OpenAI, 2026)

Fixing one finite polynomial prevents the gradient exponent from drifting

The formal no-opening conclusion is converted, through the source’s finite-dimensional formal and analytic inputs, into a power bound

$$|x{|}^m\le C_N(|\nabla f_N(b,x)|+|x{|}^{N-1}),$$

where $x$ consists of opening coordinates and $f_N$ is one finite Taylor polynomial of the action. The exponent $m$ is fixed before $N$ is chosen. Take $N-1>m$ and shrink the neighborhood so the remainder can be absorbed whenever required by the subsequent comparison.

The ordinary analytic gradient inequality is then applied to that one fixed polynomial. Localization errors may subsequently decrease as radii grow without changing its exponent. Rechoosing an unrelated Taylor truncation at every radius would not supply a single exponent for the flow-length argument. The formal algebraic input is used only through this finite-power conclusion here. (OpenAI, 2026)

Delayed radii turn earlier energy drops into summable future errors

Let $a_j$ measure the square root of the Gaussian energy drop on the $j$th time interval. Good intervals are controlled by a concave power of the energy, whose decrements telescope. On bad intervals the localization estimate gives powers of earlier drops, schematically

$$a_j\le e^{-c{q}^j{R}_{\mathrm{in}}^2}+\sum _{i<j}{a}_i^{\kappa q^{j-i}},\kappa >1,q>1.$$

If all $a_i\le \epsilon <1$, then each power is bounded by $a_i{\epsilon }^{\kappa q^{j-i}-1}$. Summing over future offsets gives a coefficient tending to zero with $\epsilon$. Choose it below one half and absorb the resulting multiple of the finite-prefix sum $\sum a_i$. The separate Gaussian term is summable as well.

Using finite prefixes is crucial: the estimate closes the graphical bootstrap before knowing that the graph exists for all future times. Delayed radii depend only on available past drops, so they remain legitimate at a hypothetical first failure. Finite accumulated motion prevents that failure and makes the fixed-frame core graph Cauchy. The original extracting sequence identifies its limit, while propagation and increasing radii extend convergence to every rescaled time and every compact spatial region. This controls the axis as well as the shrinker shape.

🏷️ Dependencies and related notes

The proof uses established multiplicity-one and end-structure theory, local Brakke regularity, graphical pseudolocality, positive-time smoothing, and finite-dimensional analytic tools. Its distinctive chain is the nonzero opening obstruction, a single finite Taylor truncation, a localized gradient inequality, and summability with radii constrained by propagation.

Scalar curvature and Ricci-flow singularities concerns a different geometric evolution and the question of smooth extension. Here the endpoint is uniqueness of the geometry seen by magnifying a fixed singularity. Affine Bernstein rigidity also converts global geometric control into a rigidity statement through elliptic estimates, but its convex graphs and affine area are different objects.

The review covered the theorem, geometric input, selected stationary-family and lateral-action arguments, the scalar nonvanishing and phase mechanisms, the finite-jet interface, localization, and the delayed-radius convergence proof. It did not independently verify every contour, gauge, inverse, and nonlinear estimate in the long stationary construction, or reprove the cited external theorems. No numerical or formal certificate was executed for this family.

🏷️ Manuscripts and verification scope

• Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow — pinned manuscript and companion source.

No family scope page was present at lean/docs/355.md in the pinned snapshot. This does not establish that all related formalization code is absent. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Uniqueness of tangent flows at the first singular time of embedded surface mean-curvature flow.