De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach

Abstract: We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

• The paper provides the first unconditional existence proof for De Giorgi varifold weak solutions to mean curvature flow.
• It introduces a novel minimizing movement scheme using a non-degenerate L² proxy combined with Mullins–Sekerka-type regularization.
• The method extends naturally to volume-preserving flows and offers a solid foundation for numerical approximations in complex geometric settings.

De Giorgi Varifold Solutions to Mean Curvature Flow via Minimizing Movements

Introduction and Background

The paper "De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach" (2607.03930) addresses the longstanding challenge of constructing global weak solutions to mean curvature flow (MCF) and its volume-preserving variant using variational time-discretization schemes. Classical approaches to MCF model the evolution of interfaces as an $L^2$-gradient flow of the perimeter functional. However, the natural $L^2$ geodesic distance between interfaces is degenerate—any two sets are at zero distance—limiting the direct application of gradient flow theory in metric spaces.

Previous pivotal works, such as Almgren-Taylor-Wang (ATW) and Luckhaus-Sturzenhecker (LS) schemes, introduced alternative proxies for measuring interfacial motion, but these approaches required restrictive regularity assumptions to establish convergence and were only capable of proving conditional existence results. These gaps have persisted especially for non-smooth flows or in the global-in-time regime, and for volume-preserving variants.

This paper delivers a new minimizing movement scheme for (volume-preserving) MCF based on a non-degenerate, robust approximation of the $L^2$ distance, leveraging a Mullins–Sekerka-type nonlocal regularization. This construction grants, for the first time, an unconditional existence proof of weak (De Giorgi varifold) solutions via time-discrete variational methods, independent of regularity or comparison arguments.

Variational Framework and Metric Construction

The authors propose a minimizing movement scheme defined at each time-step $h > 0$ and regularization parameter $\alpha > 0$. At each step, the new configuration is chosen as the minimizer of the sum of the perimeter (or area) and a squared "distance" between phases, defined by: $$d_{\alpha, h}^2(A, B) = \int \left| \left( |\rho_h * \nabla \chi_A| + \alpha^2 (I - \Delta) \right)^{-1/2} (\chi_B - \chi_A) \right|^2 \, dx$$ where $\chi_A$ is the phase indicator of set $A$, $\rho_h$ is a mollifier at scale $h$, and $L^2$0 is the identity minus the Laplacian. This metric regularizes the degenerate $L^2$1-distance via convolution and a nonlocal term in the spirit of Mullins–Sekerka, which enables sufficient control to pass to the limit in the discrete approximation.

Key aspects:

• The regularization allows the metric tensor $L^2$2 to be expressed robustly in terms of the phase indicator functions, without requiring strict regularity of the sets involved.
• For fixed $L^2$3, the theory of gradient flows in metric spaces (Ambrosio-Gigli-Savaré framework) applies to yield existence and compactness for the discrete scheme.
• The appropriate limiting procedure is first $L^2$4 (time-step vanishing), followed by $L^2$5 (removal of regularization).

Notion of Solution and Main Theorem

The paper rigorously formulates the definition of admissible couples—evolving phase indicators and varifolds—that satisfy compatibility and measurable energy dissipation. The primary notion of solution is a De Giorgi varifold solution, characterized by:

• Admissibility: The varifold is compatible with the geometric interface determined by the evolving phase indicator.
• Square Integrable Velocity: Existence of a velocity field $L^2$6 defined w.r.t. the limiting varifold measure.
• Energy Dissipation: A sharp gradient-flow-type energy inequality

$L^2$7

holds for almost every $L^2$8, describing the balance between perimeter decay and dissipative mechanisms (generalized mean curvature, velocity).

The main theorem asserts that, for $L^2$9 and any initial set of finite perimeter, the limit (flat flow) of the minimizing movement scheme constructed with the distance $L^2$0 is a global, unconditional (i.e., no perimeter convergence assumption) De Giorgi varifold solution to (volume-preserving) MCF. For the volume-preserving case, the construction directly imposes the volume constraint without penalization.

Analysis and Convergence

The technical analysis proceeds via meticulous a priori estimates at the discrete level, ensuring monotonicity and compactness of the perimeter, uniform control of the interpolant and dissipation terms, and extracting convergent subsequences for the varifolds and velocity fields through Helly’s selection principle and Aubin-Lions-Simon compactness arguments. In the limit as the time step vanishes, the authors establish that the limiting pair satisfies the admissibility and energy inequality conditions for a De Giorgi varifold solution.

A subtle point is the handling of the nonlocal regularization parameter $L^2$1. The limit $L^2$2 is performed after the time-discretization limit, ensuring that all controls are uniform and that the essential structures are preserved. The analysis relies on results by Schätzle for chemical potentials in $L^2$3 in dimensions $L^2$4, but the authors argue that analogous results could be extended to higher dimensions with suitable generalizations.

Notably, the proof avoids any maximum principle or comparison arguments—a significant departure from approaches relying on Allen–Cahn-type approximations.

Implications and Future Directions

This work closes a crucial gap regarding the existence and convergence of minimizing movement schemes for mean curvature flow, establishing that a robust, regularized proxy for the $L^2$5 metric permits applying general gradient flow theory even in the geometric setting where the original $L^2$6 metric is degenerate.

Theoretical Implications:

• Unconditional Existence: The scheme yields global-in-time weak solutions without perimeter convergence assumptions or regularity on the initial data.
• Rigorous Justification of Minimizing Movements: The result validates that the minimizing movements interpretation of MCF, in De Giorgi's spirit, can indeed provide unconditional weak solutions.
• Volume Preservation Without Penalization: For volume-constrained flows, the proposed scheme accommodates constraints directly in the minimization, extending beyond the scope of previous penalization-based results.
• Weak-Strong Consistency: The scheme satisfies a weak-strong uniqueness principle: if the classical solution exists, the limiting object coincides with it, ensuring consistency with smooth theory.

Practical Implications:

• The variational discrete scheme provides a natural and efficient basis for numerical approximations of geometric flows, robust even when interfaces develop singularities or lose regularity.
• The approach is modular and can potentially be adapted for more complex settings, such as multi-phase flows, interface coupling with advection, capillarity, or generalizations in higher dimensions.

Prospects for Generalization:

The techniques may extend to anisotropic mean curvature flows, sharp interface limits of diffuse interface models, and interface evolution on manifolds. Substituting the Laplacian with $L^2$7-Laplacian operators for $L^2$8 (as suggested) opens the route to higher-dimensional extensions, subject to establishing corresponding compactness and regularity results for the chemical potentials.

Numerical Results and Contradictory Claims

The paper does not present explicit numerical results; its primary contribution is theoretical. Notable strong claims substantiated by the analysis include:

• First unconditional existence proof for MCF/volume-preserving MCF via minimizing movements.
• No need for maximum principle or penalization for the volume constraint.

No contradictory or contentious claims versus the prior literature are evident; instead, the work clarifies and synthesizes gaps previously encountered in minimizing movement approaches.

Conclusion

By introducing a robust, non-degenerate proxy for the $L^2$9-distance in the context of geometric variational flows, this paper establishes the first unconditional existence result for (volume-preserving) mean curvature flow via the minimizing movements framework. The methodology is purely variational and adapts the machinery of gradient flows in metric spaces to the geometric case, ultimately yielding De Giorgi varifold solutions without regularity or comparison assumptions. The approach not only resolves longstanding analytical issues but provides solid ground for further mathematical and computational developments in the theory of geometric evolution equations (2607.03930).