Explain convexity of K-energy leading to unique metrics.
problem Uniqueness of constant scalar curvature Kahler metrics and extremal metrics.
method Convexity of K-energy along weak geodesics in Kahler potentials.
result Uniqueness of extremal metrics up to automorphisms.
Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.
problem Relate the energy spectrum to the simple length spectrum of metrics on surfaces.
method Analyze the energy spectrum of metrics on surfaces and their Teichmüller spaces, considering homotopy conditions.
result The energy spectrum determines the simple length spectrum under certain conditions.
Proves positive energy conjecture for a specific metric class.
problem Proving the positive energy conjecture for a class of AHM metrics.
method Analyzes asymptotically Horowitz-Myers (AHM) metrics on R2imesTn−2. result Generalizes previous results on positive energy conjecture.
Smooth minimizers of K-energy are cscK metrics in cohomologous Kähler classes.
problem Regularity of weak minimizers of the K-energy in Kähler manifolds.
method Analyzing the extended K-energy and using J-properness.
result Finite energy minimizers are smooth cscK metrics.
Paper introduces a complete metric topology for low energy spaces.
problem Defining a topology for low energy spaces with prescribed singularity.
method Introduces a completely metrizable topology stronger than capacity convergence.
result Low energy spaces have a natural completely metrizable topology.
Developed techniques to prove uniqueness of Sasaki-extremal metrics.
problem Proving uniqueness of Sasaki-extremal metrics on Sasakian manifolds.
method Pluripotential theoretic techniques for transversally holomorphic foliations.
result Uniqueness of Sasaki-extremal metrics for fixed transversally holomorphic structures.
Geodesics found in a metric space of m-subharmonic functions.
problem Metric structure on energy class of m-subharmonic functions.
method Inspired by Kähler geometry, introduced a metric structure and studied metric convergence.
result Geodesics constructed in a subspace of the complete metric space.
Paper characterizes Kähler-Einstein metrics on log Fano pairs.
problem Characterizing Kähler-Einstein metrics on log Fano pairs.
method Introducing a geodesic metric structure on Kähler potentials and using energy properness results.
result Existence of Kähler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy.
Geodesics in non-Archimedean metrics are continuous.
problem Understanding geodesics in spaces of non-Archimedean metrics.
method Maximal psh segments are geodesics, and continuity of these segments is proven.
result Maximal psh segments joining continuous psh metrics are continuous.
We introduce and study the notion of the energy of a smooth metric measure space, which includes as special cases the Yamabe constant and Perelman's ν-entropy. We then investigate some properties the energy shares with these constants, in particular its relationship with the κ-noncollapsing property. Finally, we us…
AdS uniqueness and black hole energy bounds proven.
problem Proving uniqueness of Anti-de Sitter spacetime and energy bounds for AdS black holes.
method Adapted Wang's proof to static asymptotically locally hyperbolic vacuum metrics and higher-genus horizons.
result Negativity of free energy E−TS for AdS black holes with higher-genus horizons. Paper renormalizes volume of singular Yamabe metrics.
problem Renormalizing volume of singular Yamabe metrics.
method Volume renormalization for singular Yamabe metrics.
result Existence of a conformally invariant energy.
In this paper, we prove the equivalence of the existence of extremal Kahler metrics and the properness of the modified K energy on projective bundles. Moreover, we discuss the relations of the lower boundedness of the K energy, the infimum of the Calabi energy and the extremal polynomials. In particular, we give an exa…
Quantitative stability for nearly minimizing Yamabe metrics.
problem Understanding the stability of nearly minimizing metrics in Riemannian geometry.
method Proving quantitative closeness of nearly minimizing metrics to minimizing metrics in a specific sense.
result The distance between nearly minimizing metrics and minimizing metrics is controlled quadratically by the Yamabe energy deficit.
Study on energy of maps from K3 surface to flat orbifold.
problem Energy of maps from K3 surface to flat orbifold.
method Investigate Dirichlet energy of smooth maps and introduce an invariant.
result Ratio of energy to invariant converges to 1 for Foscolo's collapsing families.
Study finite-energy metrics over complex manifold degenerations.
problem Finite-energy metrics on complex manifolds with singularities.
method Investigate spaces of plurisubharmonic metrics with finite-energy conditions.
result Complete and geodesic metric structure on finite-energy metrics space.
Derive Dirichlet scalar curvature energy functional variation formula
problem Dirichlet scalar curvature energy functional
method First variation formula
result Introduce Dirichlet-Einstein metrics
The paper studies K-energy on compactifications of Lie groups and proves the existence of Kahler-Einstein metrics.
problem Existence of Kahler-Einstein metrics on compactifications of Lie groups.
method Criterion for K-energy properness, alternative proof of Delcroix's theorem, study of minimizers.
result Alternative proof of Delcroix's theorem for Fano manifolds.
The problem of recovering the asymptotics of a short range perturbation of the Euclidean metric on R^n from fixed energy scattering data is studied. It is shown that if two such metrics, g1, g2, have scattering data at some fixed energy which are equal up to smoothing, then there exists a diffeomorphism ψ`fixing infini…
Constructs metrics with negative constant scalar curvature.
problem Negative constant scalar curvature metrics.
method One-parameter family of complete metrics.
result Verifies positive energy conjecture for these metrics.
Using Hilbert's criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or…
The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.
problem Classifying contact 3-manifolds with critical metrics and understanding their entropy.
method Critical metrics optimization and entropy analysis.
result Anosov contact metrics' optimization is linked to Reeb dynamics and entropy.
Study on a metric space derived from Kähler manifolds.
problem Understanding the geometry of low energy classes on Kähler manifolds.
method Introduced a metric dψ on the low energy space Eψ of a Kähler manifold (X,ω). result Demonstrated that the triangle inequality holds for the metric dψ. Paper proves existence of constant scalar curvature Kähler metrics under certain conditions.
problem Existence of constant scalar curvature Kähler metrics.
method Generalized apriori estimates and used automorphism group discreteness, K-energy non-increasing, and properness of K-energy.
result Proves equivalence of non-existence of cscK metric and existence of a destabilized geodesic ray with non-increasing K-energy.
In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric…
Weyl energy decreases for connected sums of certain four-manifolds.
problem Finding metrics with minimized Weyl energy on connected sums of four-manifolds.
method Proving existence of a metric on the connected sum with strictly smaller Weyl energy than the sum of energies of the original manifolds.
result Weyl energy of the connected sum is strictly smaller than the sum of energies of the original manifolds.
The paper derives index-energy estimates for Yang-Mills connections and Einstein metrics.
problem Estimating the index of Schrödinger operators and its relation to energy.
method Conformally invariant estimates for Schrödinger operators and their application to Yang-Mills connections and Einstein metrics.
result Sharp growth rate of the index in terms of energy for Yang-Mills connections and conformally invariant estimates for Betti numbers.
We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal Kähler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.
We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …
We investigate the low-energy behavior of the gradient flow of the L2 norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.
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…
New metric defines surface shapes, minimizing area and angle distortions.
problem Defining and measuring the shape of high genus surfaces.
method Defined a metric space, introduced energies for area and angle distortions, showed minimizers by lower semicontinuity.
result Energy minimizers in surface shape space correspond to quasiconformal homeomorphisms.
We present a new method to compare the shapes of genus-zero surfaces. We introduce a measure of mutual stretching, the symmetric distortion energy, and establish the existence of a conformal diffeomorphism between any two genus-zero surfaces that minimizes this energy. We then prove that the energies of the minimizing …
Researchers introduce new energies to study constant scalar curvature metrics.
problem Understanding constant scalar curvature metrics on compact Kähler manifolds.
method Introduced a family of Kβ energies using Berman's quantization and intersection theory. Combined with non-Archimedean techniques, provided a uniform Yau-Tian-Donaldson correspondence. result Uniform Yau-Tian-Donaldson correspondence characterizes the existence of a unique constant scalar curvature Kähler metric.
The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.
problem Defining a metric for high energy spaces in Kähler manifolds.
method Endowing the high energy space with a metric that makes it a complete geodesic metric space.
result The geodesic metric space (Ep(X,θ),dp) is uniformly convex for p>1. Establishes convexity and coercivity of K-energy functional for complex tori.
problem Convexity and coercivity of K-energy functional for complex tori.
method Geodesics in finite energy space, cone angle perturbations, stability of coercivity.
result Openness of coercivity under cone angle perturbations and existence of cscK cone metrics.
The paper classifies cosymplectic manifolds with critical metrics in dimension 3.
problem Classifying cosymplectic manifolds with critical metrics.
method Study of Chern-Hamilton energy functional on compact cosymplectic manifolds.
result Classification of manifolds admitting critical compatible metrics in dimension 3.
In this paper, we discuss a Donaldson's version of the modified K-energy associated to the Calabi's extremal metrics on toric manifolds and prove the existence of the weak solution for extremal metrics in the sense of convex functions which minimizes the modified K-energy.
New metric spaces for geodesic rays in cohomology classes.
problem Constructing geodesic rays in cohomology classes with finite energy.
method Introduced a chordal metric and proved geodesic properties.
result Found a characterization of geodesic rays in terms of test curves.
Based on Donaldson's method, we prove that, for an integral Kahler class, when there is a Kahler metric of constant scalar curvature, then it minimizes the K-energy. We do not assume that the automorphism group is discrete.
Study geometrical properties of oscillator group with a Lorentzian metric.
problem Geometrical analysis of oscillator group.
method Bi-invariant Lorentzian metric, homogeneous Ricci solitons, harmonicity properties, energy functional.
result Determination of critical points for energy functional and explicit calculation of their energy.
Derives stress-energy identities in Liouville theory on compact surfaces.
problem Stress-energy tensor correlation functions on compact Riemann surfaces.
method Varying correlation functions with respect to background metric, treating different types of variations separately.
result Stress-energy correlation functions expressed as differential operators acting on primary field correlation functions.
New energy definition for expanding de Sitter spacetime with umbilic boundaries.
problem Defining energy for spacetimes with expanding de Sitter background and umbilic boundaries.
method Adapting Liu-Yau energy to a quasi-local setting in expanding de Sitter spacetime.
result Positivity of the defined energy for certain values of the cosmological constant.
We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes Eχ~(X,ω). Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of c…
New formula connects Loewner energy to moving frames' renormalised energy.
problem Calculating Loewner energy of Jordan curves.
method Using renormalised energy of moving frames.
result Loewner energy as Kähler potential for Weil-Petersson space.
Given a compact polarized Kähler manifold X↪CPN, the space of Bergman metrics on X, parameterized by SL(N+1,C), corresponds to a dense set in the space of Kähler potentials in the Kähler class as N→∞. Critical points of the kth K-energy functional, which is def…
In 1996, Shi generalized the epsilon-regularity theorem of Schoen and Uhlenbeck to energy-minimizing harmonic maps from a domain equipped with a bounded measurable Riemannian metric. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi's th…
Paper examines stability of minimizing metrics on manifolds with boundary.
problem Stability of minimizing Yamabe metrics on compact manifolds with boundary.
method Investigates stability in the sense introduced by Escobar, showing closeness of nearly minimizing metrics.
result Quantitative closeness of nearly minimizing metrics to minimizing Yamabe metrics within their conformal class.