https://neckpinch.com/topics/free-boundary/New arXiv papers on free boundary & capillary, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.30771arxiv:2610.30771Thu, 22 Oct 2026 00:00:00 GMTJinyu Guo, Liangjun Weng — In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space. We construct a locally constrained nonlinear curvature flow that preserves the k-th quermassintegral while monotonically increasing the l-th quermassintegral for any (1≤ l<k≤ n). Using this flow, we establish a full family of the quermassintegral inequalities for horo-convex capillary hypersurfaces in a horoball with contact angle θ∈(0,π/2]. Moreover, we weaken the horo-convexity to convexity and obtain the quermassintegral inequalities for convex capillary hypersurfaces between the n-th and l-th quermassintegrals.Other curvature flowsVolume-preserving & constrained flowsFree boundary & capillaryGeometric inequalities https://arxiv.org/abs/2610.09557arxiv:2610.09557Wed, 07 Oct 2026 00:00:00 GMTYueheng Bao, Lorenzo Sarnataro — We provide an alternative and independent proof of the analogue of the Gage-Hamilton-Grayson Theorem for embedded free boundary curve shortening flows in a convex planar domain established by Langford-Zhu. Our proof is based on blowup arguments introduced by B. White in his work on mean-convex mean curvature flow.Mean curvature flowCurve shorteningSingularitiesFree boundary & capillary https://arxiv.org/abs/2610.05909arxiv:2610.05909Mon, 05 Oct 2026 00:00:00 GMTFuya Hiroi — We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles θ_±, the angle θ between the skew lines, and the initial data γ₀, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.Other curvature flowsVolume-preserving & constrained flowsStability & uniquenessBoundaries & non-compactFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2609.03185arxiv:2609.03185Wed, 02 Sep 2026 00:00:00 GMTSourav Ghosh — We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form S^(n-k) × ℝ^k_+ or S^(n-k)_+ × ℝ^k, then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2609.01026arxiv:2609.01026Tue, 01 Sep 2026 00:00:00 GMTYueheng Bao, Robert Haslhofer — In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.Mean curvature flowSingularitiesWeak & singular flowsStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2608.22574arxiv:2608.22574Sun, 23 Aug 2026 00:00:00 GMTHunter Liu, Sebastian Munoz, Zihui Zhao — We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees 0,1,2,3. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions 2≤ n ≤ 4000 is verified by a rigorous computer-assisted argument using interval arithmetic.Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessFree boundary & capillaryEigenvalues & spectrum https://arxiv.org/abs/2608.22494arxiv:2608.22494Sun, 23 Aug 2026 00:00:00 GMTXinqun Mei, Guofang Wang, Liangjun Weng — We prove the rigidity conjecture of [16, Conjecture 1.2] for smooth strictly convex capillary Gauss solitons in a Euclidean half-space with an acute contact angle: every such soliton is a spherical cap. Combined with our previous convergence result for the capillary Gauss curvature flow [16, Theorem 1.1], it follows that the flow starting from a strictly convex capillary hypersurface with an acute contact angle converges to a capillary spherical cap, after a suitable rescaling.Other curvature flowsGauss curvature flowStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2607.27711arxiv:2607.27711Thu, 30 Jul 2026 00:00:00 GMTShrey Aryan, Bang-Xian Han, Zhuo-Nan Zhu — We use inverse mean curvature flow to construct bi-Lipschitz maps that preserve normalized volume and decrease distances. These maps send a round sphere onto any smooth closed strictly convex hypersurface in a sphere and a flat disk onto any smooth strictly convex free-boundary disk in a Euclidean ball. In dimension two, this proves a conjecture of E. Milman for every smooth two-sphere with Gaussian curvature at least one and gives an analogous result for nonnegatively curved disks whose boundary has geodesic curvature one. The spherical result proves the two-dimensional case of the spectral comparison conjectured by Colding and Minicozzi. Counterexamples in dimensions n≥3 show that the restriction to dimension two is sharp. Furthermore, an equivariant extension of this construction yields, for every n≥2, a contracting transport map from the uniform probability measure on a round hemisphere to the uniform probability measure on any closed geodesically convex subset of positive volume. This settles the remaining uniform-target case of a question raised by Beck and Jerison. In dimension two, we also find geometric conditions under which the uniform measure on the hemisphere can be transported by a contracting map to a broad class of nonuniform probability measures supported on domains in a hemisphere.Mean curvature flowOther curvature flowsInverse mean curvature flowSurfacesFree boundary & capillaryEigenvalues & spectrum https://arxiv.org/abs/2607.26923arxiv:2607.26923Wed, 29 Jul 2026 00:00:00 GMTTianci Luo, Yong Wei, Rong Zhou — We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the α-Gauss curvature flow ∂ₜX=−K^αν, α>0. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If α>1/(n+2), we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.Other curvature flowsGauss curvature flowSingularitiesFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2607.00094arxiv:2607.00094Tue, 30 Jun 2026 00:00:00 GMTTokuhiro Eto, Jiwoong Jang — This paper studies a Chambolle-type minimizing movement scheme for mean curvature flow with prescribed contact angle in a smooth bounded domain. The scheme is based on the capillary functional and the geodesic signed distance relative to the container, and yields a time-discrete level-set approximation. The main result asserts that, for every Lipschitz-continuous boundary function prescribing a strictly nondegenerate contact angle, the approximate solutions converge locally uniformly to the unique viscosity solution of the corresponding level-set mean curvature equation with oblique derivative boundary condition. This improves a previous convergence theorem, where the container was assumed to be convex and a curvature-type condition relating the tangential derivative of the prescribed contact-angle function to the principal curvatures of the container boundary was imposed. The main new ingredient is a uniform Lipschitz estimate for the solutions of the variational problems defining the scheme. This estimate is derived by applying a Bernstein-type argument to a suitable weighted gradient, rather than to the gradient itself, which rules out boundary maxima without relying on the previous curvature-type condition.Mean curvature flowWeak & singular flowsBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2606.29058arxiv:2606.29058Sat, 27 Jun 2026 00:00:00 GMTEzequiel Barbosa, Rosivaldo Gonçalves, Luan de Figueiredo — We study a parabolic obstacle problem for surfaces evolving by anisotropic mean curvature flow subject to an obstacle constraint. Given a convex obstacle and initial data, we seek an evolving surface minimizing an anisotropic energy functional while remaining above the obstacle; as a special case, this framework includes the anisotropic Stefan problem, where the free boundary represents a phase transition interface with direction-dependent surface tension. The central tool is the Cahn–Hoffman transform S(x) = A^(−1/2)x, which maps the Wulff ellipsoid x : x^T A⁻¹x ≤ 1\ to the Euclidean unit ball and converts the anisotropic problem into an equivalent isotropic one with a generalized Robin-type condition on the free boundary. We prove optimal regularity of the solution (C^(1,α) in space and C^(0,α/2) in time up to the free boundary) and C^(1,α)-regularity of the evolving free boundary at non-degenerate points. The parabolic Hausdorff dimension of the space-time singular set is shown to be at most n − 1.Mean curvature flowAnisotropic & nonlocal flowsFree boundary & capillary https://arxiv.org/abs/2606.17353arxiv:2606.17353Mon, 15 Jun 2026 00:00:00 GMTYueheng Bao, Robert Haslhofer — We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types ℝ^k× S^(n-k), ℝ^k_+× S^(n-k) or ℝ^k× S^(n-k)_+. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.Mean curvature flowAncient solutionsSingularitiesWeak & singular flowsFree boundary & capillary https://arxiv.org/abs/2604.15993arxiv:2604.15993Fri, 17 Apr 2026 00:00:00 GMTShujing Pan, Julian Scheuer — This paper is about hypersurfaces with boundary lying in the Euclidean unit ball, which meet the unit sphere at a fixed angle θ∈(0,fracπ2]. Such hypersurfaces are called θ-capillary hypersurfaces and for those we introduce a new notion of convexity, which we call θ-horocap-convexity. For such hypersurfaces, we prove the convergence of a curvature flow of Guan/Li type with capillary boundary. Remarkably, we prove this result for a class of curvature functions which include all quotients of symmetric polynomials and, as a consequence, we obtain the full set of quermassintegral inequalities in the θ-horocap-convex case. In the strictly horocap-convex setting, we employ the flow to prove the geometric inequalities, while for the horocap-convex case and the characterization of the equality case, we develop new arguments which are interesting in their own right.Other curvature flowsFree boundary & capillaryGeometric inequalities https://arxiv.org/abs/2603.06949arxiv:2603.06949Fri, 06 Mar 2026 00:00:00 GMTTheodora Bourni, Nathan Burns, Mat Langford — We establish a sharp rate of convergence for a free-boundary curve shortening flow in a convex domain in ℝ² which converges in finite time to a round half-point.Curve shorteningSingularitiesStability & uniquenessFree boundary & capillary https://arxiv.org/abs/2602.20865arxiv:2602.20865Tue, 24 Feb 2026 00:00:00 GMTHuy The Nguyen, Artemis A. Vogiatzi — We study curve shortening flow in high codimension for arcs with free boundary meeting a fixed smooth barrier orthogonally. We prove dilation-invariant curvature and higher-derivative estimates up to the boundary using a Stahl-type localised maximum principle and an adapted cut-off. Using a reflected Gaussian entropy and blow-up analysis, Type I boundary singularities yield a shrinking semicircle model after reflection. Type II blow-ups give a Grim Reaper translator, which is ruled out under a free-boundary entropy bound <2. Hence in the low-entropy regime the flow either converges to the orthogonal chord or has only semicircle boundary singularities.Curve shorteningSelf-shrinkers & translatorsSingularitiesFree boundary & capillary https://arxiv.org/abs/2602.08850arxiv:2602.08850Mon, 09 Feb 2026 00:00:00 GMTTheodora Bourni, Giada Franz — We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.Mean curvature flowAncient solutionsStability & uniquenessFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces https://arxiv.org/abs/2602.07989arxiv:2602.07989Sun, 08 Feb 2026 00:00:00 GMTLinlin Fan, Peibiao Zhao — Wang, Weng and Xia[Math. Ann. 388 (2024), no. 2] studied a mean curvature type flow for the smooth, embedded capillary hypersurfaces with a constant contact angle θ∈(0,π) and confirmed the existence of solutions by the standard PDE theory. In the present paper, we study a fractional mean curvature flow for C^(1,1)-regular hypersurfaces with a capillary-type boundary condition and obtain the short time existence by the fixed point argument.Mean curvature flowVolume-preserving & constrained flowsAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillaryHarmonic maps & DeTurck trick https://arxiv.org/abs/2601.14659arxiv:2601.14659Wed, 21 Jan 2026 00:00:00 GMTGuanghan Li, Chenyang Liu — In this paper, we study the long-time existence and asymptotic behavior of an anisotropic capillary Gauss curvature flow. By studying this flow and proving its convergence to a stationary solution, we establish a new existence result for the capillary Orlicz-Minkowski problem without the evenness assumption, and provide a flow approach to the existence of smooth solutions.Other curvature flowsGauss curvature flowFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2512.14437arxiv:2512.14437Tue, 16 Dec 2025 00:00:00 GMTJingeon An-Lacroix, Kiichi Tashiro — It is known that there is a strong relation between the parabolic Allen–Cahn equation and the mean curvature flow, in the sense that the parabolic Allen–Cahn equation can be considered as a "diffused" mean curvature flow. In this work, we derive a forced mean curvature flow v=−H-∂_νlog |∇u|+f(u)/|∇u|, satisfied by level surfaces of any solution to the nonlinear parabolic equation ∂ₜu=Δu-f(u). Moreover, we introduce the notion of the inner gradient flow, and unify parabolic free boundary problems in the gradient flow framework. Finally, we consider the parabolic free boundary Allen–Cahn equation \{beginalignedat2 ∂ₜu&=Δu &&in \|u|<1\ |∇u|&=1/ε &&on ∂\|u|<1\, endalignedat and confirm that under reasonable assumptions, the C^α norm of the forcing term ∂_νlog|∇u| converges to zero at an algebraic rate as ε→ 0, uniformly in time. This implies that the parabolic free boundary Allen–Cahn equation converges to the mean curvature flow, uniformly (in ε and in time) in the C^(2,α) sense.Mean curvature flowVolume-preserving & constrained flowsFree boundary & capillaryPhase fields & multiphase flows https://arxiv.org/abs/2510.22146arxiv:2510.22146Sat, 25 Oct 2025 00:00:00 GMTCan Cui, Nung Kwan Yip — Over a bounded strictly convex domain in ℝⁿ with smooth boundary, we establish a priori gradient estimate for an anisotropic mean curvature flow with prescribed contact angle and Neumann boundary conditions. The estimates require careful analysis of the degeneracy property of the anisotropic mean curvature operator. As a result, for both problems, we can infer that the solutions converge to one that is translation invariant in time.Mean curvature flowAnisotropic & nonlocal flowsBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2510.22136arxiv:2510.22136Sat, 25 Oct 2025 00:00:00 GMTCan Cui, Nung Kwan Yip — In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in ℝ². We establish a prior gradient estimate for smooth solutions to this boundary value problem. The same approach can also handle Dirichlet boundary condition in ℝⁿ, n≥ 2. For both problems, we prove that the solutions converge to one that is translation invariant in time.Mean curvature flowAnisotropic & nonlocal flowsSurfacesBoundaries & non-compactFree boundary & capillary https://arxiv.org/abs/2509.19470arxiv:2509.19470Tue, 23 Sep 2025 00:00:00 GMTJiwoong Jang — We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we establish the local-in-time equi-boundedness of approximate solutions and a uniform L²-estimate of multipliers. The difficulty is handled by conducting blowup analysis at a point in contact to a spherical cap with sharp angle.Mean curvature flowVolume-preserving & constrained flowsSingularitiesWeak & singular flowsFree boundary & capillaryLong-time existence & convergence https://arxiv.org/abs/2508.05551arxiv:2508.05551Thu, 07 Aug 2025 00:00:00 GMTTristan C. Collins, Benjy Firester — We solve a general class of free boundary Monge-Ampère equations given by det D²u = λ(f(−u))/(g(u^⋆)h(∇u))χ_(u<0\) in ℝⁿ, ∇u (ℝⁿ) = P where P is a bounded convex set containing the origin, and h>0 on P. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons on toric Fano manifolds.Gauss curvature flowRicci solitonsKähler–Ricci solitonsKähler geometryLimit spaces & RCDFree boundary & capillaryEigenvalues & spectrum https://arxiv.org/abs/2507.12097arxiv:2507.12097Wed, 16 Jul 2025 00:00:00 GMTShujing Pan, Bo Yang — In this paper, we study inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball. We establish the existence and convergence results for a class of such flows. As an application, we derive a family of Alexandrov Fenchel inequalities for weakly convex hypersurfaces with free boundary.Other curvature flowsInverse mean curvature flowFree boundary & capillaryGeometric inequalities https://arxiv.org/abs/2506.09840arxiv:2506.09840Wed, 11 Jun 2025 00:00:00 GMTXinqun Mei, Guofang Wang, Liangjun Weng — In this article, we first introduce a Gauss curvature type flow for capillary hypersurfaces, which we call capillary Gauss curvature flow. We then show that the flow will shrink to a point in finite time. This is a capillary counterpart (or Robin boundary counterpart) of Firey's problem studied in [Mathematika 21 (1974), pp. 1-11] and Tso [Comm. Pure Appl. Math. 38 (1985), no. 6, 867-882]. Finally, we prove that its normalized flow converges to a soliton. This is a capillary counterpart of the result of Guan and Ni in [J. Eur. Math. Soc. 19 (2017), no. 12, 3735-3761]. The classification of solitons remains an open conjecture.Other curvature flowsGauss curvature flowSingularitiesFree boundary & capillary https://arxiv.org/abs/2504.00452arxiv:2504.00452Tue, 01 Apr 2025 00:00:00 GMTTakuya Sato — We consider the free boundary problems of degenerate elliptic equations that describe the level set formulation of the interface motion evolved by anisotropic forced mean curvature flows. The type of free boundary problems in this paper was initially studied as the first-order Hamilton-Jacobi-Isaacs equations arising in pursuit-evasion differential games and applied to the models of first-order front propagation in Soravia (1994). In this paper, we consider an extension of these free boundary problems to the second-order equations and give a deterministic game representation based on a discrete approximation scheme in Kohn and Serfaty (2006). Furthermore, we prove the comparison principle for our free boundary problems by using the framework of time-discrete games.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsFree boundary & capillary https://arxiv.org/abs/2503.20014arxiv:2503.20014Tue, 25 Mar 2025 00:00:00 GMTJiwoong Jang, Antoine Mellet — The Patlak-Keller-Segel system of equations (PKS) is a classical example of aggregation-diffusion equation. It describes the aggregation of some organisms via chemotaxis, limited by some nonlinear diffusion. It is known that for some choice of this nonlinear diffusion, the PKS model asymptotically leads to phase separation and mean-curvature driven free boundary problems. In this paper, we focus on the Elliptic-Parabolic PKS model and we obtain the first unconditional convergence result in dimension 2 and 3 towards the volume preserving mean-curvature flow. This work builds up on previous results that were obtained under the assumption that phase separation does not cause energy loss in the limit. In order to avoid this assumption, we rely on Brakke type formulation of the mean-curvature flow and a reinterpretation of the problem as an Allen-Cahn equation with a nonlocal forcing term.Mean curvature flowVolume-preserving & constrained flowsWeak & singular flowsFree boundary & capillaryPhase fields & multiphase flows https://arxiv.org/abs/2502.18455arxiv:2502.18455Tue, 25 Feb 2025 00:00:00 GMTLiam Mazurowski, Jintian Zhu — We prove the existence of compact surfaces with prescribed constant mean curvature in asymptotically flat and asymptotically hyperbolic manifolds. More precisely, let (M³,g) be an asymptotically flat manifold with scalar curvature R≥ 0. Then, for each constant c>0, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. Likewise, let (M³,g) be an asymptotically hyperbolic manifold with scalar curvature R≥ −6. Then, for each constant c>2, there exists a compact, almost-embedded, free boundary constant mean curvature surface Σ⊂ M with mean curvature c. The proof combines min-max theory with the following fact about inverse mean curvature flow which is of independent interest: for any T the inverse mean curvature flow emerging out of a point p far enough out in an asymptotically flat (or asymptotically hyperbolic) end will remain smooth for all times t∈ (-∞,T].Mean curvature flowOther curvature flowsInverse mean curvature flowScalar curvatureBoundaries & non-compactFree boundary & capillaryMinimal & CMC surfaces https://arxiv.org/abs/2501.06951arxiv:2501.06951Sun, 12 Jan 2025 00:00:00 GMTYipeng Wang — We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of T^(rtimes)-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the T^(rtimes)-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.Ricci flowScalar curvatureStability & uniquenessFree boundary & capillaryMinimal & CMC surfaces https://arxiv.org/abs/2412.10144arxiv:2412.10144Fri, 13 Dec 2024 00:00:00 GMTAlbert Chau, Ben Weinkove — The p-Laplacian evolution equation and the α-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.Other curvature flowsGauss curvature flowFree boundary & capillaryHarmonic maps & DeTurck trick