On Approximations and Homothetic Behavior in Mean Curvature Flow

Approximating Graphical Minimal Surfaces Through Mean Curvature Flow Sawyer Jacobson∗1and Jeffery Zhang†2 1Weston High School 2Wayland High School August 24, 2025 Abstract We implement an algorithm for approximating minimal surfaces using graphical mean curvature flow, leveraging the monotonically non-increasing area under such a flow. Our approach demonstrates well-established results in minimal surface theory, including the inheritance of planarity and rotational symmetry from their Dirichlet boundary data. Furthermore, we highlight the flow’s utility by estimating minimal surfaces with complicated boundary conditions, while offering visual insights into their evolution under the flow. The paper highlights the functionality of graphical mean curvature flow as both a tool for approximation and a geometric lens for studying minimal surfaces. 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. ∗ORCID:0009-0009-2775-6127 E-mail:[email protected] †ORCID:0009-0004-4551-3368 E-mail:[email protected] 1 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 non-increasing 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 [1], 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 usefulness of graphical mean curvature flow in estimating minimal surfaces, while also providing a visual representation of their evolution throughout the flow. 2 Algorithm Take a family of surfaces parameterized by M(u1, u2, s) = F(u1, u2) + s∂Fs ∂s , νν(u1, u2), 2 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 2.1 (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. Now, it was shown in [1] 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 Ω is 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 minimality1, implying the usefulness of such a flow to estimate 1A counterexample occurs when a surface yields singularities. 3 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. 3 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. 3.1 Inheritance of Planarity In [1], 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. (a) Initial Data (b) Intermediate Data (c) Final Data As expected, Mtconverged to a subset of π, with ∂Mt= Γ ⊆π, exhibiting the property shown in [1]. 3.2 Inheritance of Rotational Symmetry It was also shown in [1] that graphical minimal surfaces M ⊆ R3,a rotationally symmetric boundary Γ,induces rotational symmetry on M.It is 4 therefore interesting to simulate such cases. When the boundary Γ can 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. 3.3 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, 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. We see that the surface flows, eventually converging, giving us an estimate (a) Initial Data (b) Intermediate Data (c) Final Data of the minimal spanning surface for the given boundary condition. References [1] Jacobson, S., On Graphical Minimal Surfaces and the Homotheticity of Singular Behavior in Mean Curvature Flow, Zenodo, https: // doi. org/ 10. 5281/ zenodo. 16903054 (2025). 6