MCF-stack: Special Solutions of Mean Curvature Flow

§ 4.15 Two-convex Lagrangian graphical expanders5BCB359FC35B

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. Tsai–Tsui–Mu-Tao Wang — 2023-09-18 (first arXiv submission).
5BCB359FC35B Two-convexity here is a condition on a potential's Hessian, not hypersurface mean convexity.
9910637B58B3

(Expanders from homogeneous potentials) Let $n\geq 2$. Suppose $U_0:\mathbb R^ n\to \mathbb R$ is homogeneous of degree two, is $C^2$ away from the origin, and its Hessian eigenvalues obey

\[ \lambda _ i+\lambda _ j\geq 0,\qquad 1+\lambda _ i\lambda _ j\geq 0\quad (i\ne j), \qquad \prod _{i=1}^ n(1+\lambda _ i^2)^{-1/2}\geq \varepsilon _1>0. \]

Proposition 6.5 of [TsaiTsuiWang2025] produces the unique solution in the stated continuous-initial-data, smooth-positive-time class of

\[ u_ t=\sum _{i=1}^ n\arctan \lambda _ i(D^2u),\qquad u(x,0)=U_0(x),\qquad u(x,t)=t\, u(x/\sqrt t,1). \]

Then $L_ t=\{ (x,Du(x,t)):x\in \mathbb R^ n\} \subset \mathbb C^ n$, $t>0$, is self-expanding; $L_1$ satisfies $\vec H=X^\perp /2$. These are complete proper embedded graphs of dimension and codimension $n$, homeomorphic to $\mathbb R^ n$, with one end. Hessian bounds persist and higher derivatives obey the estimates in Theorem 5.6. The initial cone is prescribed through the potential; no stronger asymptotic rate is asserted here.

For a concrete nonquadratic seed, take $U_0(0)=0$ and

\[ U_0(x)=-\frac{x_1^2}{20}+\frac14\sum _{j=2}^ nx_ j^2 +10^{-4}\frac{x_1^4}{|x|^2}\quad (x\ne 0). \]

A direct differentiation bounds the operator norm of $D^2(x_1^4/|x|^2)$ by $38$. Thus one Hessian eigenvalue lies within $0.0038$ of $-0.1$, and the others within $0.0038$ of $0.5$. Both pair inequalities are strictly positive. This verifies the theorem's hypotheses while keeping the potential nonconvex. This formula describes initial data, not the positive-time expander. The illustration shows its exact initial gradient cone and marks the subsequent PDE smoothing as a construction step.

The gradient cone of the nonquadratic potential specified in the text, projected from C². It is initial data at t = 0; the positive-time expander is constructed by PDE and is not rendered here.
Exact initial-cone projection. The gradient cone of the nonquadratic potential specified in the text, projected from C². It is initial data at t = 0; the positive-time expander is constructed by PDE and is not rendered here. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian expander graph in C² · Euclidean space ℝ⁴ · Embedded
Ambient space
Euclidean space ℝ⁴
Topology
R²
Embeddedness
Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness
Noncompact
Surface genus
0 (plane topology).
Topological ends
One end of Rⁿ, n ≥ 2.
Entropy
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Lagrangian expander graph in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded
Ambient space
Euclidean dimension ≥ 5
Topology
Rⁿ
Embeddedness
Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness
Noncompact
Surface genus
Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends
One end of Rⁿ, n ≥ 2.
Entropy
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area
Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

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References on this page

  1. [TsaiTsuiWang2025] Chung-Jun Tsai, Mao-Pei Tsui, and Mu-Tao Wang. Entire solutions of two-convex lagrangian mean curvature flows. 2025. Reviewed arXiv edition 2309.09432v2. Definition 2.1; Theorem 5.6; Theorem 6.2 and Propositions 6.3/6.5. Concrete nonquadratic seed is an independent substitution into Proposition 6.5. URL: https://arxiv.org/abs/2309.09432v2.All uses in this project ↩ 1
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