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In this paper, we provide new necessary and sufficient conditions for the existence of Kähler–Einstein metrics on small deformations of a Fano Kähler–Einstein manifold. We also show that the Weil–Petersson metric can be approximated by the Ricci curvatures of the canonical L^2 metrics on the direct image bundles. In addition, we describe the plurisubharmonicity of the energy functional of harmonic maps on the Kuranishi space of the deformation of compact Kähler–Einstein manifolds of general type.

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström (2018) and coupled Kähler-Ricci solitons by Hultgren (2017), we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren (2017).

We obtain a residue formula for an obstruction to the existence of coupled Kähler–Einstein metrics described by Futaki–Zhang. We apply it to an example studied separately by Futaki and Hultgren which is a toric Fano manifold with reductive automorphism, does not admit a Kähler–Einstein metric but still admits coupled Kähler–Einstein metrics.

We study analytic properties of harmonic maps from Riemannian polyhedra into CAT(κ) spaces for κ∈{0,1}. Locally, on each top-dimensional face of the domain, this amounts to studying harmonic maps from smooth domains into CAT(κ) spaces. We compute a target variation formula that captures the curvature bound in the target, and use it to prove an Lp Liouville-type theorem for harmonic maps from admissible polyhedra into convex CAT(κ) spaces. Another consequence we derive from the target variation formula is the Eells–Sampson Bochner formula for CAT(1) targets.

Let φ∈C^0∩W_{1,2}(Σ,X) where Σ is a compact Riemann surface, X is a compact locally CAT(1) space, and W_{1,2}(Σ,X) is defined as in Korevaar-Schoen. We use the technique of harmonic replacement to prove that either there exists a harmonic map u: Σ → X homotopic to φ or there exists a nontrivial conformal harmonic map v: S^2 → X. To complete the argument, we prove compactness for energy minimizers and a removable singularity theorem for conformal harmonic maps.

We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the (n−2)-skeleton, we improve the regularity to locally Lipschitz. Finally, for points x ∈ X^{(k)} with k ≤ n−2, we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of x ∈ X.

We show that every complete nontrivial gradient Yamabe soliton admits a special global warped product structure with a one-dimensional base. Based on this, we obtain a general classification theorem for complete nontrivial locally conformally flat gradient Yamabe solitons.

In this paper we survey certain aspects of the classical Weil-Petersson metric and its generalizations. Being a natural L^2 metric on the parameter space of a family of complex manifolds or holomorphic vector bundles which admit some canonical metrics, the Weil-Petersson metric is well defined when the automorphism group of each fiber is discrete and the curvature of the Weil-Petersson metric can be computed via certain integrals over each fiber. We will discuss the case when these fibers have continuous automorphism groups. We also discuss the relation between the Weil-Petersson metric and energy of harmonic maps.