We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby …
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48 results for “singular extremals”
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…
Study of harmonic maps with extreme Kerr-like singularities.
problem Analyzing harmonic maps with specific singularities.
method Asymptotic analysis of harmonic maps from 3D Euclidean space to hyperbolic plane.
result Existence and classification of tangent harmonic maps at extreme black hole horizons.
Study on extremal subsets in geodesically complete spaces with curvature constraints.
problem Characterizing extremal subsets in GCBA spaces.
method Introduced and analyzed extremal subsets in GCBA spaces, proving their properties.
result Set of topological singularities forms an extremal subset under additional assumptions.
Unique extremal Kähler metric found near a divisor.
problem Uniqueness of extremal Kähler metric near a smooth divisor.
method Analyzes Poincaré type extremal Kähler metric with cusp singularity.
result Uniqueness of extremal Kähler metric up to holomorphic transformations.
Proves Goh conditions for singular curves with specific properties.
problem Finding optimal paths with specific geometric constraints.
method Proof of Goh conditions of order n and open mapping theorem.
result Establishes conditions for strictly singular curves of corank 1.
We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological eq…
Study classifies special metrics on specific surfaces.
problem Classifying extremal Kähler metrics with singularities.
method Analyzes Hessian of the Curvature of the Metric on K-surfaces.
result Identifies non-CSC HCMU metrics on and .
Proves invariance of weighted extremal Kähler metrics under smooth blowups.
problem Invariance of weighted extremal Kähler metrics under smooth blowups.
method Uniform coercivity estimate for the (relative, weighted) Mabuchi energy on blowups.
result Invariance of weighted extremal Kähler metrics under smooth blowups.
Systolically extremal nonpositively curved surfaces are flat with finitely many singularitiesmath.DG
The regularity of systolically extremal surfaces is a notoriously difficult problem already discussed by M. Gromov in 1983, who proposed an argument toward the existence of -extremizers exploiting the theory of -regularity developed by P. A. White and others by the 1950s. We propose to study the problem of syst…
An HCMU metric is a conformal metric which has a finite number of singularities on a compact Riemann surface and satisfies the equation of the extremal Kähler metric. In this paper, we give a necessary and sufficient condition for the existence of a kind of HCMU metrics which has both cusp singularities and conical sin…
Stability of weighted extremal manifolds proven through blowups.
problem Stability of weighted extremal manifolds.
method Blowup technique to analyze weighted extremal Kähler manifolds.
result Proves weighted extremal manifolds are relatively weighted K-polystable.
A compact manifold is called Bieberbach if it carries a flat Riemannian metric. Bieberbach manifolds satisfy an isosystolic inequality by a general and fundamental result of M. Gromov. In dimension 3, there exist four classes of non-orientable Bieberbach manifolds up to an affine diffeomorphism. In this paper, We prove…
We prove an optimal systolic inequality for nonpositively curved Dyck's surfaces. The extremal surface is flat with eight conical singularities, six of angle theta and two of angle 9pi - theta, for a suitable theta with cos(theta) in Q(sqrt{19}). Relying on some delicate capacity estimates, we also show that the extrem…
The zoology of singularities for Lorentzian manifold is slightly more complicated than for Riemannian manifolds. Our present work study Cauchy-compact globally hyperbolic singular flat spacetimes with extreme BTZ-like singular lines. We use the notion of BTZ-extension of a singular spacetime introduced in a previous pa…
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of , which imply asymptotic behavior of the solutions at i…
We consider axisymmetric stationary dirty black holes with regular non-extremal or extremal horizons, and compute their on-horizon Petrov types. The Petrov type (PT) in the frame of the observer crossing the horizon can be different from that formally obtained in the usual (but singular in the horizon limit) frame of a…
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
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…
We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.
Study on weighted cscK metrics on Kähler varieties with singularities.
problem Existence of singular weighted cscK metrics on Kähler varieties.
method Resolution of singularities, coercive weighted Mabuchi functional, construction of examples.
result Existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive.
We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extre…
The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.
problem Classifying extremal Kähler metrics on complex manifolds.
method Analyzing polynomial zeros in Calabi's extremal equation.
result No invariant complete extremal Kähler metrics on with positive bisectional curvature.
The paper proves extremal black holes form at a critical point of gravitational collapse.
problem Formation of extremal black holes in gravitational collapse.
method Constructing smooth families of spherically symmetric solutions to the Einstein-Maxwell-Vlasov system.
result Extremal Reissner-Nordström black holes form at the critical collapse threshold.
This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that…
Extending BTZ models to complete hyperbolic surfaces.
problem Extending BTZ models to complete hyperbolic surfaces.
method Proving a parametrization result for globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
result The tangent bundle of the Teichmüller space parametrizes globally hyperbolic Cauchy-maximal and Cauchy-compact locally Minkowski manifolds with extreme BTZ.
A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, wh…
Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.
problem Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
method Using moving frames to demonstrate the impossibility of isometric minimal immersion.
result Non-CSC HCMU metrics cannot be isometrically immersed into 3D space forms.
We establish a regularity result for the metric on any 4-dimensional extremal Kähler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity $L^2(|\Riem|)$ in a surrounding ball is sufficiently small compared to the pointwise n…
The paper extends a geometric model using singular curves.
problem Understanding abnormal extremals in sub-Riemannian geometry.
method Analysis of singular curves and construction of a graded Lie algebra.
result A nilpotent graded Lie algebra is constructed isomorphic to .
We investigate the geometry and topology of extremal domains in a manifold with negative sectional curvature. An extremal domain is a domain that supports a positive solution to an overdetermined elliptic problem (OEP for short). We consider two types of OEPs. First, we study narrow properties of such domains in a Hada…
Researchers investigate extremal eigenvalues of GJMS operators in fixed conformal classes.
problem Investigating extremal eigenvalues of GJMS operators in fixed conformal classes.
method Developed a general framework for existence theory of extremals, defined and investigated generalised eigenvalues, and established semi-continuity results and Euler-Lagrange equations.
result Proved several new (non)-existence results for extremals of renormalised eigenvalues over the conformal class.
The paper finds conical higher cscK metrics on minimal ruled surfaces with conical singularities.
problem Existence of conical higher cscK metrics on minimal ruled surfaces.
method Develop conical singularities along at least one of the two special divisors and use the momentum construction.
result Conical higher cscK metrics exist in each Kähler class on minimal ruled surfaces.
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.
problem Solving foliation singularities on Sasakian 5-manifolds.
method Applying the Sasaki-Ricci flow to resolve cyclic quotient foliation singularities.
result Proves a Sasaki analogue of the analytic minimal model program.
This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics a…
The intrinsic geometry of the Kerr ergosurface on constant Boyer-Lindquist (BL), Kerr, and Doran time slices is characterized. Unlike the BL slice, which had been previously studied, the other slices (i) do not have conical singularities at the poles (except the Doran slice in the extremal limit), (ii) have finite pola…
The paper studies phase transitions in random matrices and tensor unfolding for detecting signals.
problem Phase transitions in singular values and vectors of large random matrices.
method Analysis of singular values and vectors of long rectangular random matrices, and tensor unfolding algorithm for asymmetric rank-one spiked tensor models.
result An exact threshold for tensor unfolding to detect signals, independent of unfolding procedure.
In this paper, we study the possibility of inferring early warning indicators (EWIs) for periods of extreme bitcoin price volatility using features obtained from Bitcoin daily transaction graphs. We infer the low-dimensional representations of transaction graphs in the time period from 2012 to 2017 using Bitcoin blockc…
Theorem proves minimal hypersurfaces in nonnegative scalar curvature manifolds are smooth.
problem Minimal hypersurfaces with singularities in manifolds of nonnegative scalar curvature.
method Singularity removal rigidity theorems, spectral PMT for AF manifolds.
result Smoothness of minimal hypersurfaces in nonnegative scalar curvature manifolds.
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of desc…
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.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
Study delta invariant of minimal generic curves on rational surfaces.
problem Recover delta invariant of curve germs from surface singularity topology.
method Explicit formulae for minimal generic curves on rational surfaces, proving delta invariant values for quotient singularities.
result Explicit formulae and values for delta invariant of minimal generic curves on rational surfaces.
Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of …
We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons have also been found. All the examples are left-invaria…
In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…