On Approximations and Homethetic Behavior in Mean Curvature Flow

Parabola Volume 62, Issue 2 (2026) On Approximations and Homethetic Behavior in Mean Curvature Flow Sawyer Jacobson1and Jeffery Zhang2 1 Introduction Geometric flows are a class of processes that describe the evolution of geometric objects in time. These flows are often driven by intrinsic or extrinsic characteristics, and are often used to simplify complex geometric shapes. One of the most studied geometric flows is the mean curvature flow, where the time evolution of the surface at each point is given by ∂Mt ∂t ⊥=Hν, where His the mean curvature—given by half the trace of the shape operator—and νis an evolving Gauss map. It is shown in 2 that such an evolution yields a monotone nonincreasing area, implying that the surfaces tend toward minimality—a stationary point of the first variation of area—under the mean curvature flow. Thus, it becomes natural to explore the use of mean curvature flow as a numerical tool for approximating minimal surfaces, which are solutions to the classic Plateau’s problem–surfaces that can be parametrized by M(x, y, f(x, y)), with a graphically defined Dirichlet boundary condition. While theoretical underpinnings for mean curvature flow are well-established, analytical solutions often remain intractable due to the nature of non-linear partial differential equations. However, for graphically defined surfaces, mean curvature flow simplifies to a non-linear parabolic partial differential equation, suggesting numerical approximation as a useful tool for estimating solutions. This paper references the work of Sawyer Jacobson in [2], by presenting the derived algorithm for estimating graphical minimal surfaces through the use of mean curvature flow. We then provide simulations, demonstrating the effectiveness of the numerical approach by including scenarios with planar and rotationally symmetric boundary conditions, as well as the estimation of minimal surfaces for complex boundary data– where analytical solutions would remain enigmatic. The results provide evidence for previously established results in graphical minimal surface theory are true, including the inheritance of planarity and symmetry from a boundary. They also highlight the 1Sawyer Jacobson is a junior student at Weston High School, MA, USA 2Jeffery Zhang is a junior student at Wayland High School, MA, USA 1 usefulness of graphical mean curvature flow in estimating minimal surfaces, while also providing a visual representation of their evolution throughout the flow. This algorithm, however, fails when singularities arise in the flow. In geometric flows, long-term behavior often includes singularities. Mathematicians classify these as type I or type II according to the rate at which curvature blows up. Due to the nature of infinite curvature, these points require special techniques to analyze. Thus, in section 3, by deriving and applying Huisken’s monotonicity formula as well as a Shi-type estimate, we are able to demonstrate that self-shrinkers, objects that behave according to the equation:  H=1 2⟨x, ν⟩, Serve as precise models for type I singularities. This extends the scope to which mathematicians can analyze these computationally troublesome points. 2 Algorithm Take a family of surfaces parameterized by M(u1, u2, s) = F(u1, u2)+s∂Fs ∂s , νν(u1, u2), where sis a compactly supported variational map and νis a unit normal vector. Now, this family of surfaces has a first variation of area given by d ds[M(s)]s=0 =−ZM∂Fs ∂s , νHdA, where His the mean curvature. So, we choose Fssuch that ∂Fs ∂s , ν=H(Fs), yielding the monotone non-increasing d ds[M(s)]s=0 =−ZM H2dA, with equilibrium at H= 0.This motivates our definition of mean curvature flow. Definition 1 (Mean Curvature Flow).Given a family of surfaces {Mt}, the mean curvature flow is the evolution of the surface by the system (∂Mt ∂t ⊥=Hν M0=M. 2 Now, it was shown in [2] that a minimal surface Mparameterized by M(x, y, f(x, y)) must satisfy the partial differential equation ∂ ∂x fx p1+|∇f|2!+∂ ∂y fy p1+|∇f|2!= 0. So, in pair with the Dirichlet boundary condition, we get that a graphical solution to Plateau’s problem in Ωis given by the system    div ∇f √1+|∇f|2= 0 in Ω M(x, y, t) = Γ(x, y)on ∂Ω. As our mean curvature flow is monotone non-increasing, flows will generally tend towards minimality3, implying the usefulness of such a flow to estimate minimal surfaces. In our graphical case, the mean curvature flow evolution simplifies to the system          ∂f ∂t =p1+|∇f|2div ∇f √1+|∇f|2in Ω×(0,∞) f(x, y, t) = Γ(x, y)on ∂Ω×(0,∞) f(x, y, 0) = ρ(x, y)in ¯ Ω×{0} for some function ρ(x, y):Ω−→ Rsuch that ρ= Γ on ∂Ω. 2.1 Simulations Now that we have a way to estimate graphical minimal surfaces, we aim to simulate multiple cases. All simulations and visualizations were implemented in Mathematica. 2.1.1 Inheritance of Planarity In [2], it was shown that graphical minimal surfaces M ⊆ R3with boundary Γ⊆ π, for some plane π, will be such that M⊆π. Therefore, we look to simulate such cases for Mt, t ∈[0, T ).In this simulation, we use the initial surface data f(0, x, y) := sin(πx 5) sin(πy 5).We also fix a uniformly-zero Dirichlet boundary condition on [0,5] × [0,5].Then, we let the surface flow, governed by the partial differential equation in 2. We see, as expected, a smoothing effect, as the surface tends towards its planar boundary. As expected, Mtconverged to a subset of π, with ∂Mt= Γ ⊆π, exhibiting the property shown in [2]. 3A counterexample occurs when a surface yields singularities. 3 (a) Initial Data (b) Intermediate Data (c) Final Data 2.1.2 Inheritance of Rotational Symmetry It was also shown in [2] that graphical minimal surfaces M ⊆ R3,a rotationally symmetric boundary Γ,induces rotational symmetry on M.It is therefore interesting to simulate such cases. When the boundary Γcan be embedded in a plane, this property is trivial as all planes are symmetric. So, we take the non-planar boundary data Γ=0.1(x−2.5)2−0.1(y−2.5)2+ 0.8,which represents the boundary of a hyperbolic paraboloid in R3.We also must define the initial boundary data, which we choose to be non-symmetric to exhibit how the symmetry of the boundary truly imposes such a property on M. f(0, x, y) := 0.1(x−2.5)2−0.1(y−2.5)2+ 0.8+0.25 exp −((x−3.0)2+ (y−3.2)2) +0.2 exp −((x−1.5)2+ (y−2.0)2) +0.15 exp −((x−4.0)2+ (y−1.0)2). +0.1 sin(2x) cos(1.2y)+0.05 sin(3y) cos(1.5x). After a short period of time, we see that the surface begins to evolve towards symmetry. Finally, it converges to the hyperbolic paraboloid, spanning the boundary defined above. (a) Initial Data (b) Intermediate Data (c) Final Data As expected, the surface Mexhibits two-fold rotational symmetry, just as does its boundary. 4 2.1.3 Complex Boundary Data One of the primary applications of graphical mean curvature flow is to estimate minimal surfaces with complicated boundary data. Therefore, we apply our algorithm to simulate such cases under multiple boundary conditions. So, we define a complicated boundary: 0.2 sin(2x) cos(1.5y)+0.15 sin(3y) cos(1.2x) + 0.1 exp −((x−2)2+ (y−3)2) −0.08 exp −((x−3.5)2+ (y−1.5)2)+ 0.05 sin(xy)+0.5, with initial condition: Γ = 0.2 sin(2x) cos(1.5y)+0.15 sin(3y) cos(1.2x)+0.1 exp −((x−2)2+ (y−3)2) −0.08 exp −((x−3.5)2+ (y−1.5)2)+0.05 sin(xy)+0.5+0.2 exp −((x−1.2)2+ (y−1.5)2) +0.15 exp −((x−3.5)2+ (y−3)2)+ 0.1 sin(2.5x) cos(1.8y) + 0.05 sin(3y) cos(2x) +0.1 exp −((x−2.5)2+ (y−2.5)2). This is analytically difficult to solve, highlighting the importance of our algorithm in estimating such surfaces. (a) Initial Data (b) Intermediate Data (c) Final Data We see that the surface flows, eventually converging, giving us an estimate of the minimal spanning surface for the given boundary condition. 3 Self-Shrinker Singularity Model for Mean Curvature Flow Though the graphical boundary case exhibits predictable behavior, at least in short time frames, the general case is far more complicated. One common occurrence is the emanation of a singularity. These can be classified as either type I or type II according to the rate at which they blow up, but for our purposes, we will only define the following: 5 Definition 2 (Type I Singularity).Let Tbe the time of the first singularity. A type I singularity is such that ∃C:sup Mt|h|2(x, t)≤C T−t, for t∈(0, T). Singularities do appear occasionally in graphical mean curvature flows; however, they are far more common in the general case. To analyze these complicated shapes, we often need to find a way to model them. Specifically, we aim to establish a model for type I singularities. To do so, we must first prove the following. Theorem 3 (Huisken’s Monotonicity Formula). ∂ ∂t ZMt Φ=−ZMt  H−∇⊥Φ Φ 2 ΦdVol, where Φis the backwards heat kernel. Proof. Recall the backwards heat kernel, Φ(x, t) = 1 (−4πt) n 2 e|x|2 4t, satisfying the backwards heat equation ∂Φ ∂t +divMt∇Φ + |∇⊥Φ|2 Φ= 0. Now, note that σ=Φ √−tsolves the backwards heat equation as well. So, we have ∂ ∂t + ∆Rn+1 Φ = ∂ ∂t + ∆Rn+1 √−tσ =∂ ∂t√−tσ+√−t∂ ∂t + ∆Rn+1 σ. Now since σis a solution to the backwards heat equation, this simplifies to ∂ ∂t + ∆Rn+1 Φ = σ−1 2√−t =Φ 2t.(1) Now, we decompose the induced Laplacian on the sub-manifold into the ambient Laplacian and the normal part as follows: ∂ ∂t +divMt∇Φ + |∇⊥Φ|2 Φ=∂ ∂t + ∆Rn+1 −∇2 ννΦ + |∇⊥Φ|2 Φ. 6 By (6), we attain ∂ ∂t +divMt∇Φ + |∇⊥Φ|2 Φ=Φ 2t−Hess(Φ)(ν, ν) + |∇⊥Φ|2 Φ.(2) Now, we see ∇2log Φ(ν, ν) = ∇∇Φ Φ(ν, ν) =∇2Φ(ν, ν) Φ−∇νΦ⊗∇νΦ Φ2 =∇2Φ(ν, ν) Φ−|∇⊥Φ|2 Φ2. Thus, Hess(Φ)(ν, ν) + |∇⊥Φ|2 Φ= ΦHess(log Φ)(ν, ν).(3) So, substituting (8), into (7), we obtain ∂ ∂t +divMt∇Φ + |∇⊥Φ|2 Φ=Φ 2t−ΦHess(log Φ)(ν, ν).(4) We know the form of Φ, so we calculate the logarithm of Φto be c(t) + |x|2 4t.We thus have Hess(log Φ) = Hess|x|2 4t. Since tis said to be fixed in computation, we have Hess(log Φ)(ν, ν) = 1 2t. Plugging this into (9), we see ∂ ∂t +divMt∇Φ + |∇⊥Φ|2 Φ= 0.(5) Now, it is shown in [1] that ∂ ∂t ZMt ΦdVol =ZMt∂ ∂tΦ−H2ΦdVol. We can insert a Laplacian due to the integration by parts formula like so: ∂ ∂t ZMt ΦdVol =ZMt∂ ∂t + ∆MtΦ−H2ΦdVol =ZMt ∂Φ ∂t + 2 D∇Φ, HE+divMt∇Φ−H2ΦdVol, 7 where divMt∇Φis the divergence of the background gradient4. Now, completing the square, ∂ ∂t ZMt ΦdVol =ZMt ∂Φ ∂t +divMt∇Φ−  H−∇⊥Φ Φ 2 Φ + ∇⊥Φ 2 ΦdVol. By (10), this simplifies to ∂ ∂t ZMt ΦdVol =−ZMt  H−∇⊥Φ Φ 2 ΦdVol. 2 Remark 3.1.Given Φ = c(t)|x|2 4texp |x|2 4t, we have ∇Φ=c(t)∇|x|2 4texp |x|2 4t = Φ x 2t. So, ∇⊥Φ Φ=⟨x, ν⟩ 2t. Lemma 4.Let {Mt}t∈[0,T )be a mean curvature flow with |h| ≤ Cfor t∈[0, T ).Then, ∃C1(C, T)such that |∇h| ≤ C1on [T 2, T). Proof. We look to prove this lemma through a Shi-type estimate. To start, we consider ς(t, x)=α|h|2+t|∇h|2,for some α > 0.Now, we compute ∂ ∂t −∆Rn+1 ς=α∂ ∂t −∆Rn+1 |h|2+|∇h|2+t∂ ∂t + ∆Rn+1 |∇h|2. =−2α|∇h|2+ 2α|h|4+|∇h|2+t−2|∇2h|2+ϵ, where ϵis an error term comprised of low order terms. Rearranging, ∂ ∂t −∆Rn+1 ς= 2α|h|4−2α|∇h|2+|∇h|2−2t|∇2h|2+tϵ. Now, by the assumption of a bounded curvature estimate and the work of [1] to show that the error is bounded, we have the inequality ∂ ∂t −∆Rn+1 ς≤2αC4+1−2α+tDC2|∇h|2. 4The divergence of the background gradient can be decomposed: divMt(∇Φ) = divMt∇MtΦ+∇⊥Φ= ∆MtΦ + D H, ∇ΦE. 8 Now, we pick αsuch that for our largest case, t=T, 1+TDC2 2< α. Thus, ∂ ∂t −∆Rn+1 ς≤2αC4. Now, by the parabolic maximum principle, ς(t, x)≤2αC4T+ς(0, x) ≤2αC4T+αC2. Now, recalling our definition for ς(t, x),we have t|∇h|2≤2αC4T+αC2,∀t∈[0, T). Thus, we have an a priori bound |∇h| ≤ C1(T, C),∀t∈[T 2, T). 2 Now, we are able to prove the main result. Theorem 5 (Self-shrinkers as a Model for Type I Singularities).Let {Mt}t<T ⊂Rn+1 be a smooth mean curvature flow. Suppose (x0, T)is a type I singularity. Then, for all sequences λk→ ∞,the rescaled flows M(k) s:= λkMT+λ−2 ks−x0, s < 0 sub-sequentially converge smoothly to a limit flow ˜ Ms,which satisfies the self-shrinker equation  H=1 2⟨x, ν⟩. Geometrically, this represents a shape-preserving property under uniform shrinking. Proof. Recall for a type I singularity, we have Csuch that sup Mt|h|2(x, t)≤C T−t, for t∈(0, T).After a parabolic rescaling, we have T−t=−λ−2 ks. So, |hM(k) s|=1 λk|hMt| ≤ 1 λk C √T−t=C −s,∀k. 9