A speed preserving Hilbert gradient flow for generalized integral Menger curvature

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Fluorescent Staining of Silicone Micro-and Nano-patterns for Their Optical Imaging

Performance of engineered surfaces can be enhanced by making them hydrophobic or superhydrophobic via coating them with low-surface-energy micro-and nano-patterns. However, the wetting phenomena of particularly irregular shape and spacing (super)hydrophobic patterns such as polysiloxane coatings are not yet fully understood from a microscopic perspective. Here, we show a new method to collect 3D confocal images from irregular polysiloxane micro-and nanorods from a single rod resolution to discuss their wetting response over long liquid/solid interaction times and quantify the length and diameter of these rods. To collect such 3D confocal images, fluorescent dye containing water droplets were left on our superhydrophobic and hydrophobic polysiloxane coated surfaces. Then their liquid/solid interfaces were imaged at different staining scenarios: (i) using different fluorescent dyes, (ii) when the droplets were in contact with surfaces, or (iii) after the droplets were taken away from the surface at the end of staining. Using such staining strategies, we could resolve the micro-and nanorods from root to top and determine their length and diameter, which were then found to be in good agreement with those obtained from their electron microscopy images. 3D confocal images in this paper, for the first time, present the long-time existence of more than one wetting state under the same droplet in contact with surfaces, as well as external and internal three-phase contact lines shifting and pinning. In the end, these findings were used to explain the time-dependent wetting kinetics of our surfaces. We believe that the proposed imaging strategy here will, in the future, be used to study many other irregular patterned (super)antiwetting surfaces to describe their wetting theory, which is today impossible due to the complicated surface geometry of these irregular patterns.

Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that considered a static growth potential, i.e., the initial growth potential is only advected by diffeomorphisms. We focus on the mathematical study of the corresponding system of coupled PDEs describing the joint dynamics of the diffeomorphic transformation together with the growth potential on the moving domain. Specifically, we prove the uniqueness and long time existence of solutions to this system with reasonable initial and boundary conditions as well as regularization on deformation fields. In addition, we provide a few simple simulations of this model in the case of isotropic elastic materials in 2D.

Combinatorial Ricci Flows with Applications to the Hyperbolization of Cusped 3-Manifolds

In this paper, we adopt combinatorial Ricci flow to study the existence of hyperbolic structure on cusped 3-manifolds. The long-time existence and the uniqueness for the extended combinatorial Ricci flow are proven for general pseudo 3-manifolds. We prove that the extended combinatorial Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature, and the flow converges exponentially fast in this case. For an ideally triangulated cusped 3-manifold admitting a complete hyperbolic metric, the flow provides an effective algorithm for finding the hyperbolic metric.

GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS

In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k-functional with Higgs self-interaction, we show that, provided $\dim (M)<2(k+1)$ , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis.

Orlicz–Minkowski flows

We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if they exist, solve the regular Orlicz–Minkowski problems. As an application, we obtain old and new existence results for the regular even Orlicz–Minkowski problems; the corresponding $$L_p$$ L p version is the even $$L_p$$ L p -Minkowski problem for $$p>-n-1$$ p > - n - 1 . Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz–Minkowski problems; the $$L_p$$ L p versions are the even $$L_p$$ L p -Minkowski problem for $$p>0$$ p > 0 and the $$L_p$$ L p -Minkowski problem for $$p>1$$ p > 1 . In the final section, we use a curvature flow with no global term to solve a class of $$L_p$$ L p -Christoffel–Minkowski type problems.

A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem

<p style='text-indent:20px;'>In this paper, a generalitzation of the <inline-formula><tex-math id="M2">\begin{document}$ L_{p} $\end{document}</tex-math></inline-formula>-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for <inline-formula><tex-math id="M3">\begin{document}$ c = 1 $\end{document}</tex-math></inline-formula>.</p>