We prove a version of the affine Kempf-Ness theorem for non-algebraic symplectic structures and shifted moment maps, and use it to describe hyperkahler quotients of T*G, where G is a complex reductive group.
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48 results for “Kempf-Ness type functional”
Develops infinite-dimensional Kempf-Ness theory for complexification-free groups.
problem Infinite-dimensional Kempf-Ness theorem challenges due to lack of complexification.
method Uses Cartan bundles to generalize theory, defining essential objects.
result Establishes convexity properties and generalized Futaki character.
Study on line bundle flow on Kähler surfaces converging to a singular solution.
problem Analyzing the mean curvature flow on Kähler surfaces.
method Investigates the flow under hypercritical phase and semipositivity conditions.
result The flow converges to a singular solution away from curves of negative self-intersection.
Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.
problem Non-positivity of Hirzebruch form on stable weights
method Kempf--Ness and frame-potential inequality
result Zero locus of Hirzebruch form on stable weights corresponds to flat logarithmic connections
We formulate a Calabi-Yau type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri's Calabi-Yau e…
The paper studies conditions for graphs connecting level sets of harmonic polynomials.
problem Conditions for graphs connecting level sets of harmonic polynomials.
method Algebraic properties and Kempf-Ness functional construction.
result Stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation.
This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic su…
The paper trivializes moment maps for various geometric structures.
problem Trivializing moment maps for different geometric structures.
method General framework of a reductive group acting on a smooth affine variety, using Kempf-Ness theory, Morse theory, and ideas from Nakajima and Kronheimer.
result Locally trivial fibration of moment maps over a regular locus of the center of the Lie algebra of a maximal compact subgroup.
For linear actions of real reductive Lie groups we prove the Kempf-Ness Theorem about closed orbits and the Kirwan-Ness Stratification Theorem of the null cone. Since our completely self-contained proof focuses strongly on geometric and analytic methods, essentially avoiding any deep algebraic result, it applies also t…
We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this…
Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopesmath.OC
Paper addresses optimization on Hadamard manifolds, generalizing gradient flow.
problem Optimization of convex functions on Hadamard manifolds.
method Introduces a generalized gradient flow to minimize .
result Gradient flow attains infimum in limit for basic manifolds.
We prove that constant scalar curvature Kähler metric "adjacent" to a fixed Kähler class is unique up to isomorphism. This extends the uniqueness theorem of Donaldson and Chen-Tian, and formally fits into the infinite dimensional G.I.T picture described by Donaldson. We prove that the Calabi flow near a cscK metric exi…
The purpose of this paper is to give a self-contained exposition of the Atiyah-Bott picture for the Yang-Mills equation over Riemann surfaces with an emphasis on the analogy to finite dimensional geometric invariant theory. The main motivation is to provide a careful study of the semistable and unstable orbits: This in…
Study on octonionic Nahm's equations and their moduli space properties.
problem Properties of octonionic Nahm's equations and their moduli space.
method Analyzing basic properties, constructing solutions, introducing symmetry, proving theorems.
result Moduli space of smooth solutions to octonionic Nahm's equations over [0,1] is a star-shaped smooth manifold.
Gradient descent on Hadamard manifolds converges to boundary points, solving optimization problems.
problem Optimization on Hadamard manifolds with unbounded convex functions.
method Gradient descent, duality theorem, moment-weight inequality.
result Gradient descent converges to boundary points, solving optimization problems.
Geometric invariant theory introduces stability conditions mirroring abelian category theory.
problem Stability conditions in geometric invariant theory.
method Axiomatic notion of central charge and stability condition on schemes and stacks.
result Introduction of stability conditions for polarized schemes and smooth projective varieties.
Study local perturbations of vector bundles with polynomial curvature solutions.
problem Existence and stability of solutions to geometric PDEs under deformations.
method Geometric invariant theory, moment map framework, polystability conditions.
result Existence and uniqueness of solutions under local polystability conditions.
Regularization leads to balancedness in deep linear networks.
problem Balancedness in deep linear networks.
method Geometric invariant theory and Riemannian geometry of fibers.
result Balancing flows converge to the balanced manifold at a uniform exponential rate.
Study on homological Dehn functions of groups of type .
problem Understanding the homological Dehn functions of groups of type .
method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type with quartic homological Dehn function and unsolvable word problem.
We generalize Newton-type methods for minimizing smooth functions to handle a sum of two convex functions: a smooth function and a nonsmooth function with a simple proximal mapping. We show that the resulting proximal Newton-type methods inherit the desirable convergence behavior of Newton-type methods for minimizing s…
The paper studies special surfaces with a new type of support function.
problem Characterizing surfaces with a specific quadratic support function.
method Developed a Weierstrass type representation involving holomorphic functions.
result Classified surfaces of rotation with this new type of support function.
Defines spherical type surfaces via support function and classifies them.
problem Characterizing surfaces via support function.
method Weierstrass type representation for SS-surfaces with prescribed Gauss map.
result Every compact and connected SS-surface is the sphere.
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted as the equations governing families of critical points of functions obeying the classical linear Darboux equations for conjugate nets.The distinguished role of the Euler-Poisson-Darboux equations and associated Lauricella-type functions is em…
This paper presents novel mixed-type Bayesian optimization (BO) algorithms to accelerate the optimization of a target objective function by exploiting correlated auxiliary information of binary type that can be more cheaply obtained, such as in policy search for reinforcement learning and hyperparameter tuning of machi…
Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.
problem Understanding the homotopy type of stabilizers of smooth functions on surfaces.
method Analyzing the homotopy properties of stabilizers for a specific class of smooth functions.
result The homotopy type of the connected component of the identity map of the stabilizer is completely described for Morse-Bott functions.
We investigate the Hawking energy of small surfaces in space times without symmetry assumptions by introducing the notion of Hawking type functionals. In particular, we find that Hawking type functionals are generalized Willmore functionals which allows us to find area constrained, minimizing, immersed, haunted bubble …
We introduce the coupled Ricci-Calabi functional and the coupled H-functional which measure how far from a coupled Kähler-Einstein metric in the sense of Hultgren-Witt Nyström. We first give corresponding moment weight type inequalities which estimate each functional in terms of algebraic invariants. Secondly, we give …
Improved understanding of translating solitons using new techniques.
problem Understanding translating solitons in geometry.
method Using a new test function and gradient estimate technique.
result Better Bernstein type result of translating solitons.
This paper is devoted to the study of non-existence of certain type of convex functions on a Riemannian manifold with a pole. To this end, we have developed the notion of odd and even function on a Riemannian manifold with a pole and proved the non-existence of non-trivial and non-negative differentiable odd convex fun…
Uniform estimates of Green functions and Sobolev-type inequalities on real and complex manifoldsmath.CV
The study establishes inequalities for functions on manifolds using Green function estimates.
problem Developing inequalities for functions on manifolds.
method Used integral representations and uniform estimates for Green functions.
result Proved Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds.
Ultraknots and limit knotsmath.GT
The abstract proves every knot type can be parametrized by smooth functions and studies limit knot types.
problem Understanding all knot types and their parametrizations.
method Proving every knot type can be represented by smooth functions and studying limit knot types.
result Every knot type can be parametrized by smooth functions and limit knot types exist.
New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
problem Vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
method Utilized refined Kato type inequalities and Böchner technique to generalize results to -integrable pluriharmonic functions and harmonic 1-forms.
result Proved vanishing property of pluriharmonic functions with finite energy on complete Kähler manifolds.
Researchers generalize Ribaucour-type surfaces with new mathematical representation.
problem Defining and characterizing new geometric surfaces.
method Developed a new mathematical representation for GRT-surfaces involving holomorphic functions and a real function.
result Explicit examples and classification of GRT-surfaces of rotation.
In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space of strongly pseudo-convex complex Finsler metrics on -- a holomorphic vector bundle over a closed Kähler manifold . This Donaldson type functional is a generalization in the complex…
Let be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various -related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…
A new GP framework for discovering unknown functions and hypergraph structure.
problem Discovering unknown functions and hypergraph structure in data.
method Interpretable Gaussian Process framework for Type 3 problems.
result Polynomial complexity for data-driven discovery of unknown functions and hypergraph structure.
The article proves a new entropy formula for surfaces with boundaries.
problem Entropy formula for surfaces with boundaries.
method Established a monotonicity formula of Hamilton type entropy.
result Entropy functional and -functional relation studied.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
In this paper we define a new cohomology of a smooth manifold called Lichnerowicz type cohomology attached to a function. Firstly, we study some basic properties of this cohomology as: a de Rham type isomorphism, dependence on the function, singular forms, relative cohomology, Mayer-Vietoris sequence, homotopy invarian…
We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions, which we call generalized self-concordant functions. This notion allows us to develop a unified framework for …
Study examines surfaces with bounded fractional mean curvature, proving control over local parametrization.
problem Understanding surfaces with bounded fractional mean curvature.
method Investigates bounded L^p-norm of fractional mean curvature, proving control over local parametrization.
result Proves control over local parametrization, leading to lower Ahlfors-regularity, weak Michael-Simon type inequality, and stability application.
We study the spectral functions, and in particular the zeta function, associated to a class of sequences of complex numbers, called of spectral type. We investigate the decomposability of the zeta function associated to a double sequence with respect to some simple sequence, and we provide a technique for obtaining the…
Paper characterizes embeddability of function spaces into -type RKBS via metric entropy.
problem Characterizing embeddability of function spaces into -type RKBS.
method Establishes a connection between metric entropy growth and embeddability.
result A bound on metric entropy growth allows embedding into -type RKBS.
Alternative proofs for various inequalities on Riemannian manifolds.
problem Various functional inequalities on Riemannian manifolds.
method Generic functional inequality, Riccati pairs, solving Riccati-type ODE.
result Alternative proofs for multiple inequalities, including Hardy-type and Caccioppoli inequalities.
The paper improves energy decay estimates for Dir-stationary Q-valued functions and applies them to Liouville-type theorems and continuity.
problem Improving energy decay estimates for Dir-stationary Q-valued functions.
method Establishing improved decay estimates and applying them to derive Liouville-type theorems and continuity.
result Dir-stationary Q-valued functions exhibit the Lebesgue property and reside in a generalized Campanato-Morrey space.
A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…