Generating shrinkers by mean curvature flow: Dedicated to Tom Ilmanen (1961–2025)

If T_pos < ∞, then there is a shrinker at time T_pos of genus ≥ 1.

“Theorem 7. If T pos < ∞, then there is a shrinker at time T pos of genus ≥ 1.”

For a set Q of rays as specified in Theorem 12, there exists a smoothly embedded shrinker Σ that has the symmetries of Q and is disjoint from Q.

“(1) Σ has the symmetries of Q and is disjoint from Q.”

For large g, there are shrinkers that have the defining properties of those non-existent minimal surfaces.

“By contrast, we show that (for large g) there are shrinkers that have the defining properties of those non-existent minimal surfaces.”

If T_fat < ∞, then there is a shrinker at time T_fat with genus ≥ 1.

“Theorem 5.If T fat < ∞, then there is shrinker at time T fat with genus ≥ 1.”

• cites [CHH22]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [BK23]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If (p,t) is a singular point with t ≤ T_fat, and if there is an ε > 0 such that genus(M(τ) ∩ B(p,ε)) ≤ g for every regular time τ ∈ (t-ε,t), then (p,t) has genus ≤ g.

“Theorem 3. If (p, t) is a singular point with t ≤ T fat , and if there is an ϵ > 0 such that genus(M (τ ) ∩ B(p, ϵ)) ≤ g for every regular time τ ∈ (t -ϵ, t), then (p, t) has genus ≤ g.”

Section 7 produces shrinkers that, for large genus, look like the union of a sphere and a bisecting plane, with the circle of intersection replaced by necks.

“Section 7 produces shrinkers that, for large genus, look like the union of a sphere and a bisecting plane, with the circle of intersection replaced by necks.”

If M ∈ A and t < min{T_Q, T_pos} is a regular time of the flow M(•), then M(t) ∈ A.

“Theorem 15.If M ∈ A and if t < min{T Q , T pos } is a regular time of the flow M (•), then M (t) ∈ A .”

Section 8 produces new examples that are shrinker analogs of some well-known, non-existent minimal surfaces.

“Section 8 produces new examples that are shrinker analogs of some well-known, non-existent minimal surfaces.”

If K and M are disjoint compact sets, and no connected component of M^c contains points of K and of Q, then T_Q(M) < T_Q(K).

“Lemma 13.Suppose K and M are disjoint compact sets, and that no connected component of M c contains points of K and of Q.Then T Q (M ) < T Q (K).”

• cites [Whi95]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Under the Initial Regularity Theorem hypotheses, there is an ε > 0 such that the flow t ∈ [0,ε] → M(t) ∩ K is smooth.

“(1) The flow t ∈ [0, ϵ] → M (t) ∩ K is smooth.”

In the regular n-gon case, for all sufficiently large n, the shrinker must be non-compact.

“In the regular n-gon case, one can show that for all sufficiently large n, the shrinker must be non-compact.”

Under the Initial Regularity Theorem hypotheses, if M_i(t) are mean curvature flows converging weakly to M(•), then for 0 < ε' < ε, t ∈ [ε',ε] → M_i(t) ∩ K converges smoothly to t ∈ [ε',ε] → M(t) ∩ K.

“(2) if t ∈ [0, T ) → M i (t) are mean curvature flows that converge weakly to M (•), then for 0 < ϵ ′ < ϵ, t ∈ [ϵ ′ , ϵ] → M i (t) ∩ K converges smoothly to t ∈ [ϵ ′ , ϵ] → M (t) ∩ K.”

H^1(Q^c; Z_2) is isomorphic to Z_2^g.

“Thus H 1 (Q c ; Z 2 ) ∼ = Z 2 g .”

In the regular n-gon case, as n → ∞, the shrinker converges to a multiplicity 2 plane.

“Indeed, one can show that as n → ∞, the shrinker converges to a multiplicity 2 plane.”

If M_i(•) is a sequence of unit-regular integral Brakke flows converging to a unit-regular Brakke flow M(•), and ν_i is an orientation of M_i(•), then after passing to a subsequence, the ν_i(•) converge to an orientation ν(•) of M(•).

“Theorem 11.Suppose M i (•) is a sequence of unit-regular integral Brakke flows that converge to a unit-regular Brakke flow M (•).Suppose ν i is an orientation of M i (•).Then, after passing to a subsequence, the ν i (•) converge to an orientation ν(•) of M (•).”

If T ≤ T_fat, then the sum of the genuses of the singularities at time T is less than or equal to the genus of M.

“Corollary 4. Suppose T ≤ T fat .Then the sum of the genuses of the singularities at time T is less than or equal to the genus of M .”

The shrinker Σ in Theorem 12 occurs as a shrinker in a mean curvature flow M(•) in R3 for which M(0) is a compact, smoothly embedded surface in Q^c.

“(3) Σ occurs as a shrinker in a mean curvature flow M (•) in R 3 for which M (0) is a compact, smoothly embedded surface in Q c .”

If the orientations ν_i in Theorem 11 are strong, then the limiting orientation ν is strong.

“If the ν i are strong, then so is ν.”

Under the Initial Regularity Theorem hypotheses, if M_i(0) converges smoothly to M(0), then t ∈ [0,ε] → M_i(t) ∩ K converges smoothly to t ∈ [0,ε] → M(t) ∩ K.

“(3) Suppose in Assertion (2) that M i (0) converges smoothly to M (0).Then t ∈ [0, ϵ] → M i (t) ∩ K converges smoothly to t ∈ [0, ϵ] → M (t) ∩ K.”

The paper proves existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.

“We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.”

• Also stated in: § 1. Introduction · plain

  “In this paper, we produce shrinkers by creating initial surfaces that, under mean curvature flow, develop singularities at which the desired shrinkers occur as blowups.”

The convergence Σ_g → Σ is smooth at the origin.

“Claim4. The convergence Σ g → Σ is smooth at the origin.”

The family F of surfaces in M''_g constructed in the proof of Proposition 44 contains a critical surface M.

“Theorem 53.Let F be the family of surfaces in M ′′ g constructed in the proof of Proposition 44.Then F contains a critical surface M .”

• AssumptionRests on an assumption: The family F is connected.

The convergence in Theorem 42 is smooth with multiplicity 1 away from the circle {z = 0} ∩ S(2).

“The convergence is smooth with multiplicity 1 away from the circle {z = 0} ∩ S(2).”

• cites [BNS24, Theorem 1.3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the singularity of Theorem 43, the entropy of Σ_g is bounded above by a constant independent of g.

“(4) The entropy of Σ g is bounded above by a constant independent of g.”

In the singularity of Theorem 40, every shrinker Σ_g has entropy ≤ δ.

“(4) Σ g has entropy ≤ δ.”

Under the hypotheses of Lemma 25, if c is the number of components of ∂M that lie in {z > 0} and wind once around Z, and d is the number of points in M ∩ Z_+ = M ∩ Z ∩ {z > 0}, then genus(M) = (c + d)g.

“Let c be the number of components of ∂M that lie in {z > 0} and that wind once around Z. Let d be the number of points in M ∩ Z + = M ∩ Z ∩ {z > 0}. Then genus(M ) = (c + d)g.”

If Σ ∈ S_g, then Σ is G_g-invariant.

“Theorem 55. Suppose Σ ∈ S g . Then (1) Σ is G g -invariant”

As g → ∞, the shrinker Σ_g of Theorem 40 converges to the union of the plane {z = 0} and the sphere ∂B(0, 2).

“Furthermore, for large g, Σ g has exactly one end, and, as g → ∞, Σ g converges to the union of the plane {z = 0} and the sphere ∂B(0, 2).”

If K is a smooth 3-dimensional manifold-with-boundary, M = ∂K, and W is a relatively open subset of K such that M ∩ W has positive genus, then M ∩ W contains a simple closed curve that is homologically nontrivial in K.

“Lemma 19.Suppose that K is a smooth, 3-dimensional manifold-with-boundary.Let M = ∂K.Suppose that W is a relatively open subset of K such that M ∩W has positive genus.Then M ∩ W contains a simple closed curve that is homologically nontrivial in K.”

If Σ_i are shrinkers that converge weakly to a multiplicity 1 plane Σ, then Σ_i is a multiplicity 1 plane for all sufficiently large i.

“Lemma 63.Suppose Σ i are shrinkers that converge weakly to a multiplicity 1 plane Σ.Then Σ i is a multiplicity 1 plane for all sufficiently large i.”

All shrinkers at (p, T) have the same genus.

“Theorem 68.All shrinkers at (p, T ) have the same genus.”

• AssumptionRests on an assumption: The set S of all shrinkers at (p, T) is connected with respect to smooth convergence.

Under the hypotheses of Theorem 17, if Ω is a simply connected open subset of Q^c, then genus(M ∩ Ω) = 0.

“if Ω is a simply connected open subset of Q c , then genus(M ∩ Ω) = 0.”

In the singularity of Theorem 40, every shrinker Σ_g satisfies Σ_g ∩ {z = 0} = Q_g.

“(2) Σ g ∩ {z = 0} = Q g .”

If M ∈ A, then T_Q ≤ T_pos.

“Theorem 20.Suppose M ∈ A .(See Definition 14.) Then T Q ≤ T pos .”

There is a constant Λ < ∞ independent of g such that area(M[s] ∩ B(0, r))/(πr^2) < Λ for all s ∈ (0, 1) and all 0 < r < 2.

“There is a constant Λ < ∞ (independent of g) with the following property: area(M [s] ∩ B(0, r)) πr 2 < Λ for all s ∈ (0, 1) and all 0 < r < 2.”

In the proof of Theorem 40, we can choose the initial surface M so that entropy(Σ) < δ.

“In the proof of Theorem 40, we can choose the initial surface M so that entropy(Σ) < δ.”

If M ∈ M_g, T_pos < ∞, and Σ is a shrinker at the origin at time T_pos, then Σ has genus g or genus 2g.

“Furthermore, genus(Σ) ≤ genus(M )/2, and Σ has genus g or genus 2g.”

As g tends to infinity, the Σ_g of Theorem 43 converge to the plane {z = 0} with multiplicity 3.

“Furthermore, as g tends to infinity, Σ g converges to the plane {z = 0} with multiplicity 3.”

If M ∈ M_g and T_pos = ∞, then M(t) converges smoothly to {z = 0} with multiplicity 1.

“Suppose M ∈ M g .If T pos = ∞, then M (t) converges smoothly to {z = 0} with multiplicity 1.”

• cites [SW89]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [Whi05]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionRests on an assumption: The area of the evolving surface M(t) is a decreasing function of t.

If Σ ∈ S_g, then genus(Σ) is g or 2g.

“(4) genus(Σ) is g or 2g.”

For each Q as in Theorem 12, there exists a family F satisfying the hypotheses of Theorem 23.

“Theorem 24.For each Q as in Theorem 12, there exists a family F satisfying the hypotheses of Theorem 23.”

If Σ+ is nonempty, then it meets {z = 0} orthogonally.

“Furthermore, if Σ + is nonempty, then it meets {z = 0} orthogonally.”

If M_n are compact, smoothly embedded surfaces in Q^c such that T_Q(M_n) = ∞, and M_n converge smoothly to a surface M in A, then M(T_Q) ∩ Q = {0}, where T_Q = T_Q(M).

“Theorem 22. Suppose M n are compact, smoothly embedded surfaces in Q c such that T Q (M n ) = ∞.Suppose the M n converge smoothly to a surface M in A .(See Definition 14.) Then M (T Q ) ∩ Q = {0}, where T Q = T Q (M ).”

If T_pos < ∞, then the origin is singular at time T_pos.

“If T pos < ∞, then the origin is singular at time T pos , and if Σ is a shrinker at (O, T pos ), then Σ has genus > 0.”

Under the hypotheses of Lemma 60, Γ_g divides Σ_g into two components.

“Then Γ g divides Σ g into two components.”

The origin is a regular point of the flow M(•) at all times < T_pos.

“The origin is a regular point of the flow M (•) at all times < T pos .”

If B is a ball centered at a point p in the x-axis X, and Σ_g ∩ B converges smoothly with some multiplicity m to {z = 0} ∩ B, then m is odd.

“Claim 2. Suppose that B is a ball centered at a point p in the x-axis, X, and that Σ g ∩ B converges smoothly with some multiplicity m to {z = 0} ∩ B. Then m is odd.”

For large g, Σ_g has exactly one end.

“For large g, Σ g has exactly one end.”

• cites [BNS24, Theorem 1.3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If M ∈ M_g, T_pos < ∞, and Σ is a shrinker at the origin at time T_pos, then genus(Σ) ≤ genus(M)/2.

“Furthermore, genus(Σ) ≤ genus(M )/2, and Σ has genus g or genus 2g.”

• AssumptionRests on an assumption: The singularities at O and O_∞ at time T have the same genus, namely genus(Σ).

Suppose M ∈ A, T := T_Q(M) < ∞, p ∈ M(T) ∩ Q, and Σ is a shrinker at (p,T). If p = 0, then Σ is a connected, G-invariant, genus-g surface disjoint from Q.

“(2) If p = 0, then Σ is a connected, G-invariant, genus-g surface disjoint from Q.”

As g → ∞, the Σ_g converge to the union of the plane {z = 0} and the sphere S(2).

“As g → ∞, the Σ g converge to the union of the plane {z = 0} and the sphere S(2).”

• cites [BNS24, Theorem 1.3]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

If in Theorem 61 the Σ_g converge to the plane {z = 0} with multiplicity 3, then for all sufficiently large g, Σ_g has exactly 3 ends.

“Proposition 62. Suppose, in Theorem 61, that the Σ g converge to the plane {z = 0} with multiplicity 3. Then all sufficiently large g, Σ g has exactly 3 ends.”

Suppose M ∈ A, T := T_Q(M) < ∞, p ∈ M(T) ∩ Q, and Σ is a shrinker at (p,T). If p ≠ 0, then Σ is a cylinder whose axis is the line {rp : r ∈ R}.

“(1) If p ̸ = 0, then Σ is a cylinder whose axis is the line {rp : r ∈ R}.”

If M ∈ M_g and T_pos = ∞, then there is a time T < ∞ such that M(t) is a smooth graph for all t ≥ T.

“In particular, there is a time T < ∞ such that M (t) is a smooth graph (i.e., such that M (t) projects diffeomorphically onto {z = 0}) for all t ≥ T .”

The set U_g is an open subset of M''_g with respect to smooth convergence.

“Lemma 51.The set U g is an open subset of M ′′ g with respect to smooth convergence.”

• AssumptionRests on an assumption: Smooth graphs never develop singularities.

The points in C are regular points of the flow at all times t ∈ [0, T_pos].

“The points in C are regular points of the flow at all times t ∈ [0, T pos ].”

• cites [Whi21, Theorem 15.1]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In the singularity of Theorem 43, every shrinker Σ_g has genus 2g and 3 ends.

“(1) Σ g has genus 2g and 3 ends.”

After passing to a subsequence, the convergence in Theorem 43 is smooth away from one circle.

“After passing to a subsequence, the convergence is smooth away from one circle.”

Let U be a G-invariant region in R^3 such that ∂U is smooth and U is diffeomorphic to an open ball. If M ⊂ U is a smoothly embedded, G-invariant, compact, connected 2-manifold-with-boundary such that ∂M = M ∩ ∂U, M ∩ {z = 0} ∩ U = Q ∩ U, and genus(M) ≤ 2g, then M has genus 0, g, or 2g.

“Let U be a G-invariant region in R 3 such that ∂U is smooth and such that U is diffeomorphic to an open ball.Suppose that M ⊂ U is a smoothly embedded, G-invariant, compact, connected 2-manifold-with-boundary such that ∂M = M ∩ ∂U, M ∩ {z = 0} ∩ U = Q ∩ U,and genus(M ) ≤ 2g. Then M has genus 0, g, or 2g.”

The family F ∩ U_g is nonempty: there exist M ∈ F for which T_pos(M) = ∞.

“Claim 1.The family F ∩ U g is nonempty: there exist M ∈ F for which T pos (M ) = ∞.”

If M is a G_g-invariant embedded shrinker in R^3 such that M ∩ {z = 0} = Q_g and genus(M) ≤ 2g, then M has genus g or 2g.

“Suppose M is a G g -invariant embedded shrinker in R 3 such that M ∩ {z = 0} = Q g and such that genus(M ) ≤ 2g.Then M has genus g or 2g.”

At time T_pos(M) < ∞, the flow M(t) has a singularity at the origin with positive genus.

“By Lemma 38, we know that at time T pos (M ) < ∞, the flow M (t) has a singularity at the origin with positive genus.”

If M ∈ M_g, T_pos < ∞, and Σ is a shrinker at the origin at time T_pos, then Σ ∩ {z = 0} = Q.

“Suppose that M ∈ M g and that T pos < ∞.Let Σ be a shrinker at the origin at time T pos .Then Σ ∩ {z = 0} = Q.”

If Σ ∈ S, r > R, and ∂B(0, r) intersects Σ transversely, then genus(Σ ∩ B(0, r)) = genus(Σ).

“Corollary 67.If Σ ∈ S , if r > R, and if ∂B(0, r) intersects Σ transversely, then genus(Σ ∩ B(0, r)) = genus(Σ).”

If Σ ∈ S_g, then Σ ∩ {z = 0} = Q_g.

“(3) Σ ∩ {z = 0} = Q g”

Let M be a compact, G-invariant surface embedded in R3, disjoint from Q, and of genus ≤ g. If M has a connected component Σ whose convex hull contains points of Q, then Σ is a G-invariant, genus g surface.

“Then Σ is a G-invariant, genus g surface”

If T_pos < ∞ and Σ is a shrinker at (O, T_pos), then Σ has genus > 0.

“If T pos < ∞, then the origin is singular at time T pos , and if Σ is a shrinker at (O, T pos ), then Σ has genus > 0.”

For all sufficiently large g, there is a compact, smoothly embedded surface in S^2 × R that, under mean curvature flow, has a singularity at which every shrinker Σ_g has the properties listed in Theorem 43.

“For all sufficiently large g, there is a compact, smoothly embedded surface in S 2 × R that, under mean curvature flow, has a singularity at which every shrinker Σ g has the following properties:”

If Σ ∈ S_g, then the entropy of Σ is bounded above by a constant independent of g.

“(5) The entropy of Σ is bounded above by a constant independent of g.”

Given Λ < ∞, there exists R < ∞ such that if Σ is an embedded shrinker in R^3 with entropy(Σ) ≤ Λ and r > R, then Σ ∩ B(0, r) is connected.

“and if r > R, then Σ ∩ B(0, r) is connected.”

• AssumptionRests on an assumption: Every smooth shrinker without boundary is an unstable minimal surface with respect to the shrinker metric.

The convergence in Theorem 40 is smooth with multiplicity 1 away from the intersection {z = 0} ∩ ∂B(0, 2).

“The convergence is smooth with multiplicity 1 away from the intersection {z = 0} ∩ ∂B(0, 2).”

For t ∈ [0, T_pos], M(t) ∩ {z = 0} = Q.

“For t ∈ [0, T pos ], M (t) ∩ {z = 0} = Q.”

Suppose for each g that Σ_g ∈ S_g. Then for all sufficiently large g, the genus of Σ_g is 2g.

“Theorem 59. Suppose, for each g, that Σ g ∈ S g .Then, for all sufficiently large g, the genus of Σ g is 2g.”

The class M′′_g is nonempty.

“The class M ′′ g is nonempty.”

If M ∈ M′′_g and we give M the orientation such that ν_M(2, 0, 0) = e_3, then ν_{M(t)}(0, 0, 0) = e_3 for all t ∈ [0, T_pos(M)).

“Furthermore, if we give M the orientation such that ν M (2, 0, 0) = e 3 , then ν M (t) (0, 0, 0) = e 3 for all t ∈ [0, T pos (M )), where the orientation is the standard induced orientation as defined in Section 3.”

For large g, the shrinker Σ_g of Theorem 40 has exactly one end.

“Furthermore, for large g, Σ g has exactly one end, and, as g → ∞, Σ g converges to the union of the plane {z = 0} and the sphere ∂B(0, 2).”

Under the hypotheses of Lemma 64, the convergence M_i → M is uniform at infinity: if λ_i → 0, then λ_i M_i converges smoothly with multiplicity 1 on compact subsets of R^3 ∖ {0} to C ∖ {0}.

“Lemma 64 (Ends Lemma).Suppose 0 < R < ∞.Suppose that C is a cone that is smooth except at the origin, and suppose that M is a smooth shrinker with boundary in B(0, R) and that M is smoothly asymptotic (with multiplicity one) to C. Suppose for each i that M i is a smooth shrinker with boundary in B(0, R).Suppose that M i converges smoothly (and with multiplicity one) to M on compact subsets of R 3”

If M ∈ Cone′(Λ), then M(t) ∈ Cone′(Λ) for all t ∈ [0, T_pos].

“Also, if M ∈ Cone ′ (Λ), then M (t) ∈ Cone ′ (Λ) for all t ∈ [0, T pos ].”

Suppose g ≥ 1 is an integer and 1 + d_2 < δ < 1 + d_1. There is a compact, smoothly embedded surface in S^2 × R that, under mean curvature flow, has a singularity at which every shrinker Σ_g has the properties listed in Theorem 40.

“Suppose that g ≥ 1 is an integer and that 1 + d 2 < δ < 1 + d 1 .There is a compact, smoothly embedded surface in S 2 ×R that, under mean curvature flow, has a singularity at which every shrinker Σ g has the following properties:”

If Σ ∈ S_g, then there is a G_g-invariant, special Brakke flow t ∈ R → M'(t) in R^3 such that M'(t) = |t|^{-1/2} Σ for all t < 0.

“Lemma 58.Suppose Σ ∈ S g .Then there is a G g -invariant, special Brakke flow t ∈ R → M ′ (t) in R 3 such that M ′ (t) = |t| -1/2 Σ for all t < 0.”

If M ∈ M_g, then M(t) ∈ M_g for every regular time t < T_pos(M).

“Suppose M ∈ M g .Then M (t) ∈ M g for every regular time t < T pos (M ).”

• AssumptionRests on an assumption: The area of the evolving surface M(t) is a decreasing function of t.

If M is a surface in M_g and M′ is the connected component of M containing Q, then genus(M) = genus(M′) = 2dg, where d is the number of points in M′ ∩ Z_+.

“Suppose that M is a surface in M g and let M ′ be the connected component of M containing Q. Then genus(M ) = genus(M ′ ) = 2dg, where d is the number of points in M ′ ∩ Z + .”

If M ∈ A, then there is a connected component Σ of M such that the convex hull of Σ contains points of Q.

“Theorem 16.If M ∈ A , then there is a connected component Σ of M such that the convex hull of Σ contains points of Q.”

Let M be as in Theorem 17 and R > 0. If the convex hull of a connected component of M ∩ B(0,R) contains points in Q, then for every r > R, M ∩ B(0,r) has genus 0 or genus g.

“Theorem 18.Let M be as in Theorem 17 and let R > 0. Suppose the convex hull of a connected component of M ∩ B(0, R) contains points in Q.Then for every r > R, M ∩ B(0, r) has genus 0 or genus g.”

Lee and Zhao show that any shrinker that satisfies a natural and rather mild hypothesis does indeed arise as a blowup of mean curvature flow of an initially smooth, compact surface.

“This last defect has been somewhat overcome by recent work of Tang-Kai Lee and Xinrui Zhao [LZ24], who show that any shrinker that satisfies a natural and rather mild hypothesis does indeed arise as such a blowup.”

• cites [LZ24]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Conjecture 56 is true if g is sufficiently large by Theorem 59.

“The conjecture is true if g is sufficiently large by Theorem 59 below.”