arXiv research

Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.

Study of four-dimensional Lorentzian manifolds with real Killing spinors.

We study a family of fermionic extensions of the Camassa-Holm equation. Within this family we identify three interesting classes: (a) equations, which are inherently hamiltonian, describing geodesic flow with respect to an H^1 metric on the group of superconformal transformations in two dimensions, (b) equations which …

We address some aspects of four dimensional chiral N=1 supersymmetric theories on which the scalar manifold is described by Kähler geometry and can further be viewed as Kähler-Ricci soliton generating a one-parameter family of Kähler geometries. All couplings and solutions, namely the BPS domain walls and their supersy…

Geometric flows help solve the swampland problem by preserving Einstein equations.

In a rigorous construction of the path integral for supersymmetric quantum mechanics on a Riemann manifold, based on Bär and Pfäffle's use of piecewise geodesic paths, the kernel of the time evolution operator is the heat kernel for the Laplacian on forms. The path integral is approximated by the integral of a form on …

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E…

Study shows continuous evolution of curves in Fréchet distance.

The paper studies geometric constants under modified Ricci flows with variable parameters.

Study hexagonal network evolution under curvature flow.

Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, evolution equation of the reduced $hh$-curvature and the Ricci scalar along the Finslerian Ricci flow is obtained and it is proved that the Ricci flow preserves positivity of reduc…

Graph neural network using Beltrami flow for feature and topology evolution.

Continuous curve evolution depends on initial shape on sphere.

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems [S. Vacaru: J. Math. Phys. \textbf{49} (2008) 043504 \& Rep. Math. Phys. \textbf{63} (2009) 95] is extended to include geometric mechanics and gravity models on Lie algebroids. We prove that such evolution scenarios of geometri…

We analyse second order (in Riemann curvature) geometric flows (un-normalised) on locally homogeneous three manifolds and look for specific features through the solutions (analytic whereever possible, otherwise numerical) of the evolution equations. Several novelties appear in the context of scale factor evolution, fix…

We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally flat Ricci solitons.

The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange and Riemann geometries. We verify some assertions made in the first partner pap…

We study the evolution of the Whitney sphere along the Lagrangian mean curvature flow. We show that equivariant Lagrangian spheres in $\mathbb{C}^n$ satisfying mild geometric assumptions collapse to a point in finite time and the tangent flows converge to a Lagrangian plane with multiplicity two.

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the succes…

In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We particularly investigate the global existence of the evolution of convex hypers…

Anisotropic curvature flow studied for planar networks.

The paper examines conditions for Ricci solitons to be Ricci flat or Einstein.

The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary p…

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimension…

We give a sufficient condition ensuring that the mean curvature flow commutes with a Riemannian submersion and we use this result to create new examples of evolution by mean curvature flow. In particular we consider evolution of pinched submanifolds of the sphere, of the complex projective space, of the Heisenberg grou…

Study geometric mKdV flows for Legendrian curves in a 3-sphere.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

The study analyzes the evolution of Gaussian measures under a specific gradient flow.

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing p…

Paper studies heat flow for maps on manifolds, avoiding singularities.

In this work we illustrate some well-known facts about the evolution of $S^3$ under the Ricci flow. The Dirac flow we introduce allows us to describe the 4- dimensional metrics with constant curvature. Another new flow leads to the Eguchi-Hanson metric and can be defined either on metric or on corresponding contact for…

We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find the maximal existence time for the flow in terms of the initial data. We invest…

We consider the evolution of an almost Hermitian metric by the $(1,1)$ part of its Chern-Ricci form on almost complex manifolds. This is an evolution equation first studied by Chu and coincides with the Chern-Ricci flow if the complex structure is integrable and with the Kähler-Ricci flow if moreover the initial metric…

Lectures on surface evolution through singularities.

Study bi-harmonic flow with forcing term on smooth curves.

A new method evolves point clouds using B-splines for smooth surfaces.

Study curves evolving on hypersurfaces with free boundaries, preserving length.

New game approximates mean curvature flow evolution.

We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for stu…

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of t…

Space curves with convex projections evolve smoothly until shrinking to a point.

We describe the evolution under the mean curvature flow of embedded Lagrangian spherical surfaces in the complex Euclidean plane $\mathbb{C}^2$. In particular, we answer the Question 4.7 addressed in [Ne10b] by A. Neves about finding out a condition on a starting Lagrangian torus in $\mathbb{C}^2$ such that the corresp…

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and internal energy terms. As the mean curvature of the hypersurface is the main drivi…

We consider bosonic supersymmetric backgrounds of ten-dimensional conformal supergravity. Up to local conformal isometry, we classify the maximally supersymmetric backgrounds, determine their conformal symmetry superalgebras and show how they arise as near-horizon geometries of certain half-BPS backgrounds or as a plan…

We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we derive a general existence, uniqueness and regularity theorem for solutions of Dirac…

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain n…

The Cheeger constant increases under Ricci flow on spheres.