arXiv research

The paper decomposes metrics on manifolds with boundaries.

Revisits Koiso's rigid metrics on complex projective spaces.

This note analyzes the normal form of gradient Ricci 4-solitons.

By extending Koiso's examples to the non-compact case, we construct complete gradient Kahler-Ricci solitons of various types on certain holomorphic line bundles over compact Kahler-Einstein manifolds. Moreover, a uniformization result on steady gradient Kahler-Ricci solitons with non-negative Ricci curvature is obtaine…

We consider the non-trivial Ricci soliton on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$ constructed by Koiso and Cao. It is a Kähler metric invariant by the $U(2)$ action on $\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$. We study its Yamabe equation and prove it has exactly one $U(2)-$invariant solution up to homothecies.

Study on complex Grassmannians' rigidity using Einstein deformations.

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita…

We study infinitesimal Einstein deformations on compact flat manifolds and on product manifolds. Moreover, we prove refinements of results by Koiso and Bourguignon which yield obstructions on the existence of infinitesimal Einstein deformations under certain curvature conditions.

Let (M,g) be a compact oriented Einstein 4-manifold. Write R-plus for the part of the curvature operator of g which acts on self-dual 2-forms. We prove that if R-plus is negative definite then g is locally rigid: any other Einstein metric near to g is isometric to it. This is a chiral generalisation of Koiso's Theorem,…

The paper proves stability for a modified Bach flow on various manifolds.

Study second order integrability of Einstein deformations on Riemannian and Kähler manifolds.

In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…

In this note, stimulated by the existence result of Futaki-Ono-Wang for toric Sasaki-Einstein metrics, we obtain new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of P^1-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here the …

There are several types of equation of motion of elastic wires. In this paper, we treat an equation taking account of the thickness of wire. The equation was introduced by Caflisch and Maddocks on plane curves, and they proved the existence of solutions. Koiso and Sugimoto generalized the result to any dimensional Eucl…

English translation of "Solitony Ricciego" (Wiadomości Matematyczne 48, 2012, no. 1, pp. 1-32). Despite the general-sounding title, the text covers just a few narrow topics: Perelman's proof of the fact that compact Ricci solitons are of the gradient type, and a detailed unified description of Page's and Berard Bergery…

Establishes metrics with positive curvature on projective line bundles.

Given a compact Fano Kähler manifold (M,J) with a Kähler Ricci soliton g, we consider smooth families {J_t} of complex deformations of (M,J) which are invariant under the action of a maximal torus T in the full isometry group of (M,g). We prove that, under a certain condition on the spectrum of the Laplacian of g, ther…

Study on stability of Einstein metrics on symmetric spaces.

Rigidity of Fubini-Study metric on odd complex Grassmannians.

We study a class of asymptotically cylindrical Ricci-flat Kähler metrics arising on quasiprojective manifolds. Using the Calabi--Yau geometry and analysis and the Kodaira--Kuranishi--Spencer theory and building up on results of N.Koiso for the case of compact manifolds, we show that under rather general hypotheses any …

Study shows unique Einstein metrics on $SU_{2n+1}$ and related spaces.

Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.

We prove that the quasi-Einstein metrics found by Lü, Page and Pope on $\mathbb{C}P^{1}$-bundles over Fano Kähler-Einstein bases are conformally Kähler and that the Kähler class of the conformal metric is a multiple of the first Chern class. A detailed study of the lowest-dimensional example of such metrics on $\mathbb…

On a manifold of dimension at least six, let $(g,τ)$ be a pair consisting of a Kähler metric g which is locally Kähler irreducible, and a nonconstant smooth function $τ$. Off the zero set of $τ$, if the metric $\hat{g}=g/τ^2$ is a gradient Ricci soliton which has soliton function $1/τ$, we show that $\hat{g}$ is Kähler…

We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso--Cao soliton and the Lü--Page--Pope quasi-Einstein …

Novel analysis of generalized Ricci solitons leads to stability results and deformations.

Constructs complete metrics and solitons on complex vector bundles.

Study optimal partition problem for Q-curvature equations on Einstein manifolds.

Global convergence proved for Gursky-Malchiodi $Q$-curvature flow in dimensions $n \geq 5$.

Researchers describe and compare decompositions of Poincaré duality pairs.

The paper proposes and discusses semiorthogonal decompositions for moduli spaces of vector bundles.

New examples of Kähler-Ricci solitons on Fano threefolds with non-trivial moduli found.

The paper classifies decompositions of 3-sphere and lens spaces with handlebodies.

We combine aspects of the notions of finite decomposition complexity and asymptotic property C into a notion that we call finite APC-decomposition complexity. Any space with finite decomposition complexity has finite APC-decomposition complexity and any space with asymptotic property C has finite APC-decomposition comp…

This paper generalizes octahedral decomposition to links in thickened surfaces.

Researchers compute Goeritz groups for all (1,1)-link decompositions.

Study concordance of decompositions from defining sequences in 3-sphere.

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…

Study shows OAT decomposition generates unexplained profit and loss, while SU decompositions depend on risk factor order.

A new algorithm speeds up CP decomposition for large tensors.

Paper characterizes optimization landscape of Tucker decomposition.

A double pants decomposition of a 2-dimensional surface is a collection of two pants decomposition of this surface introduced in arXiv:1005.0073v2. There are two natural operations acting on double pants decompositions: flips and handle twists. It is shown in arXiv:1005.0073v2 that the groupoid generated by flips and h…

Smooth 4-manifolds have simple horizontal decompositions.

Let $\mathcal{J}^1$ be the real form of a complex simple Jordan algebra such that the automorphism group is $\mathrm{F}_{4(-20)}$. By using some orbit types of $\mathrm{F}_{4(-20)}$ on $\mathcal{J}^1$, for $\mathrm{F}_{4(-20)}$, explicitly, we give the Iwasawa decomposition, the Oshima--Sekiguchi's $K_ε-$Iwasawa decomp…

We study the topological types of pants decompositions of a surface by associating to any pants decomposition $P,$ in a natural way its pants decomposition graph, $Γ(P).$ This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…

New method uses random decompositions for high-dimensional Bayesian optimization.

New varifold example shows decomposition failure.

Derive new Euler-Ramanujan-type identities and infinite decompositions for zero mean curvature graphs in various spaces.