arXiv research

Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.

Solves geodesic equations on specific metrics types.

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler…

The study explores metrics with constant curvature on compact manifolds.

Study rigidifies Einstein-type manifolds with boundary and constant curvature.

We prove that the isoperimetric constant is positive for all symmetric spaces of noncompact type and compute it explicitly.

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

In this article, we introduce an analogous problem to Yamabe type problem considered by Case, J., which generalizes the Escobar-Riemann mapping problem for smooth metric measure spaces with boundary. The last problem will be called Escobar-Riemann mapping type problem. For this purpose, we consider the generalization o…

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.

We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.

Estimates eigenvalues and spectrum for graph substructures using isocapacitary constants.

In the work of Ammann, Dahl and Humbert it has turned out that the Yamabe invariant on closed manifolds is a bordism invariant below a certain threshold constant. A similar result holds for a spinorial analogon. These threshold constants are characterized through Yamabe-type equations on products of spheres with rescal…

In the theory of finite type submanifolds, null 2-type submanifolds are the most simple ones, besides 1-type submanifolds (cf. e.g., [3, 12]). In particular, the classification problems of null 2-type hypersurfaces are quite interesting and of fundamentally important. In this paper, we prove that every $δ$(3)-ideal nul…

Constructs metrics with negative constant scalar curvature.

We show that one-sided Alexandrov embedded constant mean curvature cylinders of finite type in the 3-sphere are surfaces of revolution. This confirms a conjecture by Pinkall and Sterling that the only embedded constant mean curvature tori in the 3-sphere are rotational.

Stable generalized complex structures on certain surfaces are constant.

Proves log-concavity of cluster algebra coefficients for type $A_n$.

Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.

We give a condition for an almost constant-type manifold to be a constant-type manifold, and holomorphic and $R$-invariant submanifolds of almost Hermitian manifolds are studied. Generalizations of some results in [5] are given.

Consider a compact Kähler manifold X with a simple normal crossing divisor D, and define Poincaré type metrics on X\D as Kähler metrics on X\D with cusp singularities along D. We prove that the existence of a constant scalar curvature (resp. an extremal) Poincaré type Kähler metric on X\D implies the existence of a con…

Study Cheeger inequalities for Riemannian manifolds with boundary.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

New metrics found on orbifold resolutions with specific curvature.

The paper classifies surfaces with constant curvature in 3D De Sitter and anti De Sitter spaces.

In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space $ł^3$ and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in $ł^3$ of constant mean c…

The paper proves constant rank theorems for special Lagrangian equations.

The paper studies constant mean curvature hypersurfaces in Finsler manifolds.

The paper develops quantitative estimates for holomorphic sections over bounded domains.

Solves geodesic equations on specific metrics.

Total five different types of translation surfaces, based upon planarity of translating curves and the absolute figure, arise in a Galilean 3-space. Excepting the type in which both of translating curves are non-planar we obtain these surfaces with arbitrary constant Gaussian and mean curvature.

In this paper we show explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-riemannian manifolds with constant sectional curvature. In particular, we prove that every h in [-1,-2 sqrt{n-1}/n) can be realized as the constant curvature of a complet…

We construct a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with lightlike axis and call these surfaces meridian surfaces of parabolic type. They are analogous to the meridian surfaces of elliptic or hyperbolic type. Using the i…

New mass-type invariants for cosmological space-times.

New types of Delaunay hypersurfaces found in spheres.

New complete hypersurfaces found in affine geometry.

We derive a correspondence between (Lorentzian) harmonic maps into the pseudosphere $S_1^2$, with appropriate regularity conditions, and certain connection 1-forms. To these harmonic maps, we associate a representation of type Weierstrass, and we apply it to construct timelike surfaces with constant mean curvature.

Study spherically symmetric Finsler metrics with specific curvature properties.

We show that there are isometrically nonequivalent Robertson-Walker metrics which have the same set of geodesics. While one of these metrics satisfies the Einstein equations of pure dust without a cosmological constant, all the other describe pure dust with additional energy momentum tensor of cosmological constant typ…

Proves product metrics are Yamabe metrics under small flat torus conditions.

In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, re…

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blow-up of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blow-ups of extremal Kähler surfaces

In this paper we produce families of complete non compact Riemannian metrics with positive constant $σ_k$-curvature by performing the connected sum of a finite number of given $n$-dimensional Delaunay type solutions, provided $2 \leq 2k < n$. The problem is equivalent to solve a second order fully nonlinear elliptic eq…

Global well-posedness and asymptotic convergence for vacuum Einstein's equations proved.

Study Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.

In this paper, we consider gradient estimates for two type of nonlinear parabolic equations under the Ricci flow: one is the equation $u_t=Δu+au\log u+bu$ with $a,b$ two real constants, the other is $u_t=Δu+λu^α$ with $λ,α$ two real constants. By a suitable scaling for the above two equations, we obtain Hamilton-So…

In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0,…