arXiv research

Proves existence and uniqueness of weighted metrics for smooth spaces.

The study finds positive Einstein metrics on complex manifolds and spheres.

Paper proves existence of weighted constant scalar curvature metrics.

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

Proves existence of weighted-cscK metrics on Kähler manifolds.

Existence of metrics on non-Kähler varieties, generalizing previous work.

Extremal metrics exist if uniformly $K$-stable over models.

Study on Kähler metrics on ruled surfaces, proving existence and non-existence.

The paper explores existence and non-existence of constant scalar curvature and extremal Sasaki metrics.

No Einstein metrics found on certain double disk bundles.

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…

Study of harmonic maps on 2D simplicial complexes, proving existence and regularity.

In this paper we prove the existence of coupled Kähler-Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the pro…

Study proves existence of Kähler-Einstein metrics and Ricci flat Kähler metrics in 4-manifolds.

Existence of Kähler-Einstein metrics on compactifications of Lie groups.

We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.

Investigates admissible metrics on compact Kähler varieties and their stability.

Existence and uniqueness of discrete Einstein metrics on trees proven.

Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…

In this paper we compute the Futaki invariant of adiabatic Kaehler classes on resolutions of Kaehler orbifolds with isolated singularities. Combined with previous existence results of extremal metrics by Arezzo-Lena-Mazzieri, this gives a number of new existence and non-existence results for cscK metrics.

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determi…

Study on existence of harmonic metrics for non-Hermitian Yang-Mills bundles.

Global obstructions found for conformally Einstein metrics in 6D.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Characterizes metrics on Lie groups, proving non-simultaneous existence of balanced and pluriclosed metrics.

We show that on a compact Riemannian manifold with boundary there exists $u \in C^{\infty}(M)$ such that, $u_{|\partial M} \equiv 0$ and $u$ solves the $σ_k$-Ricci problem. In the case $k = n$ the metric has negative Ricci curvature. Furthermore, we show the existence of a complete conformally related metric on the int…

Investigates special metrics in hypercomplex geometry.

Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.

Researchers prove existence of metrics maximizing Laplace eigenvalue on all closed surfaces.

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…

This paper investigates the question of which smooth compact 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL; Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples. Using twistor m…

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible sub-representations. In this situation, one can have equivalent sub-modules, leading …

Study local curvature estimates and existence of conformal metrics on noncompact manifolds.

Study Ricci-Deturck flow from rough metrics, proving short-time existence.

We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.

Proves existence of static vacuum metrics with specific boundary data.

Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle $2π(1-α)$ for $α\in (0, 1)$. In this paper we study how the existence of such Kähler-Einstein metrics depends on $α$. We show that in the negative s…

We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on $\Bbb C ^n$ without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete $U(n)$-invariant Kähler metric with non-n…

We investigate the relation between holomorphic torus actions on complex manifolds of LCK type and the existence of special LCK metrics. We show that if the group of biholomorphisms of such a manifold $(M,J)$ contains a non-real compact torus, then there exists a Vaisman metric on the manifold. Moreover, we show that i…

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor $W$ to the setting of smooth metric measure space…

Paper proves non-existence of certain balanced metrics on six-manifolds.

Hermitian metrics with zero second Chern Ricci curvature are rigid and exist on specific manifolds.

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

Uniform K-stability ensures existence of special metrics on toric manifolds.

Researchers found a way to create a special metric with a specific curvature function.

Suppose that there exist two Kähler metrics $ω$ and $α$ such that the metric contraction of $α$ with respect to $ω$ is constant, i.e. $Λ_ω α= \text{const}$. We prove that for all large enough $R>0$ there exists a twisted constant scalar curvature Kähler metric $ω'$ in the cohomology class $[ ω]$, satisfying $S(ω' ) - R…

In this paper, we study Mabuchi metrics on Fano manifolds. We prove that Mabuchi metrics exist if the modified Ding functional is proper modulo a reductive subgroup of its automorphism group. On the other hand, the inverse that Mabuchi metrics implies the properness is obtained by using Darvas-Rubinstein's properness p…