arXiv research

New approach uses isotropic geometry to solve Euclidean problems.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

Study proves only origin-centered spheres solve certain curvature problems.

Ancient solutions to Ricci flow with isotropic curvature conditions are classified.

Paper proves uniqueness of solutions to a geometric inequality problem.

This paper classifies solutions for a specific geometric problem.

The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.

The paper proves uniqueness of solutions to curvature problems using various methods.

The paper studies hanging chains and surfaces in degenerate geometries.

This paper classifies solutions to a specific hyperbolic geometry problem.

Study confirms the uniqueness of the unit sphere for a specific geometric problem.

Study 4D solitons with specific curvature properties, proving curvature bounds and classifying solutions.

The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian. This problem in the general version remains unsolved. Here, we contribute to it w…

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are wea…

Parallel spinors help characterize G2* structures and isotropic forms.

Paper proves conjecture about Einstein metrics on manifolds with positive isotropic curvature.

Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and a fibration $Π:M\rightarrow N$. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the $Π$-HJE for $X$, whose unknown is a section $σ:N\rightarrow M$ of $Π$. The standard HJE is obtained when the …

We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literatur…

New classification for higher-dimensional shrinking Ricci solitons with positive isotropic curvature.

This research solves Plateau's problem for CRPC surfaces.

Higher-dimensional Ricci flows are shown to have unique and stable solutions.

We define a notion of isotropic surfaces in $\mathbb{O}$, i.e. on which some canonical symplectic forms vanish. Using the cross-product in $\mathbb{O}$ we define a map $ρ\colon Gr\_2(\mathbb{O})\to S^6$ from the Grassmannian of $\mathbb{O}$ to $S^6$. This allows us to associate to each surface $Σ$ of $\mathbb{O}$ a fun…

The paper examines Randers metrics with isotropic scalar curvature properties.

Paper establishes a relation between Berwald scalar curvature and S-curvature.

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.

M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ…

In this paper, we find a condition on $(α, β)$-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.

We solve the analogue of Björling's problem for Willmore surfaces via a harmonic map representation. For the umbilic-free case the problem and solution are as follows: given a real analytic curve $y_0$ in $S^3$, together with the prescription of the values of the surface normal and the dual Willmore surface along the c…

Study physical work done by isotropic vector forces along isotropic curves.

Study on uniquely determining thermal properties from boundary temperature and heat flux measurements.

Study isotropic Riemannian maps and helices along them.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

The $\hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $\hat B_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $\hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) u…

Study isotropic curves on complex quadric with geometric relations.

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…

Constructs a moment map flow for isotropic maps on surfaces.

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, chara…

Spinor representation in isotropic space via Laguerre geometry.

Fundamental solutions found for PDEs in Finsler geometry.

Developed a new concept of isometric surfaces in isotropic space.

Paper shows isotropic $E$- and $S$-curvatures are equivalent in warped Finsler metrics.

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isot…

This paper concerns conditions related to the first finite singularity time of a Ricci flow solution on a closed manifold. In particular, we provide a systematic approach to the mean value inequality method, suggested by N. Le and F. He. We also display a close connection between this method and time slice analysis of …

The paper studies Kropina metrics with a specific curvature property.

We show that for $m>n\geq 2$, there are at least two exact isotropic $n$-tori in $\mathbb{C}^m$ which are not Hamiltonian isotopic in $\mathbb{C}^m$, even though they are smoothly isotopic as isotropic $n$-tori. We apply this discovery to obtain more distinct non-exact isotropic tori in $\mathbb{C}^m$.

Study classifies zero mean curvature surfaces with planar curvature lines.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by Hélein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valu…