Aronson-B\’{e}nilan estimates for weighted porous medium equations under the geometric flow

In this paper, we study Aronson-B\’{e}nilan gradient estimates for positive solutions of weighted porous medium equations $$\partial_{t}u(x,t)=\Delta_{\phi}u^{p}(x,t),\,\,\,\,(x,t)\in M\times[0,T]$$ coupled with the geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$ on a complete measure space $(M^{n},g,e^{-\phi}dv)$. As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.

Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals

We describe the construction of CMC surfaces with symmetries in $\mathbb {S}^{3}$ S 3 and $\mathbb {R}^{3}$ ℝ 3 using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.

Regularity of the Donaldson Geometric Flow

We prove a regularity theorem for the solutions of the Donaldson geometric flow equation on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The minimal initial conditions lay in the Besov space $B^{1,p}_{2}(M, {\varLambda }^{2})$ B 2 1 , p ( M , Λ 2 ) for p > 4. The Donaldson geometric flow was introduced by Simon Donaldson in Donaldson (Asian J. Math.3, 1–16 1999). For a detailed exposition see Krom and Salamon (J. Symplectic Geom.17, 381–417 2019).

Spine and Individual Vertebrae Segmentation in Computed Tomography Images Using Geometric Flows and Shape Priors

The surgical treatment of injuries to the spine often requires the placement of pedicle screws. To prevent damage to nearby blood vessels and nerves, the individual vertebrae and their surrounding tissue must be precisely localized. To aid surgical planning in this context we present a clinically applicable geometric flow based method to segment the human spinal column from computed tomography (CT) scans. We first apply anisotropic diffusion and flux computation to mitigate the effects of region inhomogeneities and partial volume effects at vertebral boundaries in such data. The first pipeline of our segmentation approach uses a region-based geometric flow, requires only a single manually identified seed point to initiate, and runs efficiently on a multi-core central processing unit (CPU). A shape-prior formulation is employed in a separate second pipeline to segment individual vertebrae, using both region and boundary based terms to augment the initial segmentation. We validate our method on four different clinical databases, each of which has a distinct intensity distribution. Our approach obviates the need for manual segmentation, significantly reduces inter- and intra-observer differences, runs in times compatible with use in a clinical workflow, achieves Dice scores that are comparable to the state of the art, and yields precise vertebral surfaces that are well within the acceptable 2 mm mark for surgical interventions.

Learning High-Order Geometric Flow Based on The Level Set Method

Recently, the development of deep learning (DL), which has accomplished unbelievable success in many fields, especially in scientific computational fields. And almost all computational problems and physical phenomena can be described by partial differential equations (PDEs). In this work, we proposed two potential high-order geometric flows. Motivation by the physical-information neural networks (PINNs) and the traditional level set method (LSM), we have integrated deep neural networks (DNNs) and LSM to make the proposed method more robust and efficient. Also, to test the sensitivity of the system to different input data, we set up three sets of initial conditions to test the model. Furthermore, numerical experiments on different input data are implemented to demonstrate the effectiveness and superiority of the proposed models compared to the state-of-the-art approach.

Construction of a family of flat curves according to the equations of isometric grids

The article reveals an analytical description of the formation of families of orthogonal flat curved lines in the implicit form based on the analysis of the parametric equation of a flat isometric grid constructed by separating the real and imaginary parts of the function of a complex variable. This problem is due to the fact that flat isometric grids, as two families of orthogonal coordinate lines with square cells, are used in conformal mappings, for example, when drawing images on curved surfaces with the least distortion. At the same time, families of flat parallel lines are widely used in geometric modeling of heat transfer, electric fields, fluid flow, etc. There is a connection between these geometric images, which is explained by specific examples. Analytical calculations of deriving the parametric equation of an isometric grid are quite time-consuming, so they are performed in the environment of symbolic algebra Maple. For this purpose, the corresponding software of the interactive model of derivation of parametric equations of isometric grids for any initial function of a complex variable with the subsequent separation of its real and imaginary parts was created. It was found that the values of the abscissa and ordinates of the parametric equation of a flat isometric grid can be represented as explicit surface equations. For integer values of the power of the exponential function of the complex variable, the values of the abscissa and the ordinate will be represented by algebraic surfaces in the explicit form. The projections of the cross sections of the abscissa and ordinate surfaces by horizontal cutting planes on the horizontal plane form two families of curved lines, the equations of which can be obtained only implicitly. By the example of the quadratic function of a complex variable, it is proved that these families of lines are mutually perpendicular. The practical application of building a family of lines for geometric modeling of fluid flow lines that flow around the barrier in the form of a semicircle is shown. Key words: isometric grids, functions of a complex variable, families of orthogonal lines, geometric flow modeling

Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy

The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n+1 dimensions. The equation turns into a harmonic oscillator form under suitable transformations. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity.

Local existence and uniqueness of skew mean curvature flow

The Skew Mean Curvature Flow (SMCF) is a Schrödinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general-dimensional SMCF in Euclidean spaces.