Minimal surfaces in hyperbolic 4-space degenerate to core of product trees.
problem Degeneration of minimal surfaces in hyperbolic 4-space.
method Study of induced metrics and geometric interpretation of minimal surfaces.
result Limits of minimal surfaces are mixed structures and cores of product trees.
New insights on solutions to Allen-Cahn equation with degenerate minimal hypersurfaces.
problem Existence and rigidity of solutions to the Allen-Cahn equation.
method Analysis of degenerate minimal hypersurfaces as limit interfaces.
result New observations and examples of solutions to the Allen-Cahn equation.
New stabilization found in planar elasticae with degenerate diffusion.
problem Existence of local minimizers in degenerate p-elasticae. method Analysis of pinned planar p-elasticae with degenerate diffusion. result Uncountably many local minimizers with diverging energy in degenerate regime.
Study horospheres in hyperbolic 3-manifolds with degenerate ends.
problem Understanding geodesics in degenerate hyperbolic 3-manifolds.
method Analyze almost minimizing geodesics to explore horospheres.
result Identify geodesics passing through thin parts of the manifold.
This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature type functional and their degeneration is related to the sphere and torus decompo…
The paper introduces scattering-symplectic manifolds and explores their properties.
problem Understanding minimally degenerate Poisson structures on manifolds.
method Constructing scattering-symplectic spheres and gluings, computing Poisson cohomology explicitly.
result Explicit computation of Poisson cohomology and new method introduced.
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.
problem Symplectic singularities and their degenerations.
method Combining volume minimization, deformation theory, and rigidity results.
result Kaledin's conjecture confirmed for symplectic singularities.
We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold of dimension at most seven, can degenerate. Loosely speaking, our results show …
We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.
The paper defines and studies d-minimal surfaces in a degenerate semi-Euclidean space.
problem Investigating surfaces in a degenerate semi-Euclidean space with a degenerate metric.
method Defining d-minimal surfaces and proving a representation formula of Weierstrass type. result The correspondence between d-minimal surfaces in R0,2,1 and spacelike flat zero mean curvature surfaces in R14. The paper constructs minimizers for deep learning networks and analyzes their geometric structure.
problem Underparametrized deep learning networks and their minimizers.
method Direct construction of minimizers without gradient descent, considering specific settings.
result Explicit family of minimizers for the global minimum and a set of degenerate local minima.
Study on hypersurfaces with minimized distance between rulings.
problem Characterizing singularities and properties of two-ruled hypersurfaces.
method Characterization through striction curves and examination of pseudo-non-degenerate properties.
result Two-ruled hypersurfaces constructed from specific curves are pseudo-non-degenerate.
Study on curves minimizing length in a degenerate metric plane.
problem Finding curves minimizing length in a plane with a degenerate metric.
method Established sufficient conditions for existence and non-existence of minimizers, using traveling wave solutions to a bi-stable Hamiltonian system.
result Existence and non-existence of minimizers can occur, with examples provided.
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
Unique K-polystable degenerations for Fano varieties confirmed.
problem Algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
method Study of optimal degeneration problems via new functionals of real valuations.
result Confirm algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
Study on minimizing network energy in R^d, introducing degenerate elastic networks.
problem Minimizing network energy in R^d with constraints on curves and junctions.
method Characterizing limits of sequences of networks bounded in energy, providing explicit representation of the relaxed problem.
result Explicit representation of degenerate elastic networks, a new concept involving only given class properties.
Smooth functions on manifolds with degenerate singular submanifolds
problem Existence of functions with minimal number of submanifolds
method Sufficient conditions for existence
result Minimal number of submanifolds
We prove existence, uniqueness and convergence of solutions of the degenerate J-flow on Kahler surfaces. As an application, we establish the properness of the Mabuchi energy for Kahler classes in a certain subcone of the Kahler cone on minimal surfaces of general type.
Study how convex structures on surfaces change.
problem Degeneration of convex structures on surfaces.
method Analyzing bordered surfaces for convex projective structures.
result Understanding how convex structures degenerate.
Proves unique degeneration of log Fano fibration germs.
problem Stable degeneration of log Fano fibration germs.
method Introduced the H-invariant for filtrations over log Fano fibration germs and used a unique quasi-monomial valuation to achieve the degeneration.
result Unique K-polystable special degeneration of log Fano fibration germs.
In this paper we show that the convergence of complete Kahler-Einstein hypersurfaces in complex torus in the sense of Cheeger-Gromov will canonically degenerate the underlying manifolds into "pair of pants" decomposition. We also construct minimal Lagrangian tori that represent the vanishing cycles of the degeneration.
Geodesics in jet space are constructed from polynomials, with some yielding globally minimizing paths.
problem Characterize geodesics in jet space and identify those that are globally minimizing.
method Sub-Riemannian geometry, Hamilton-Jacobi equations, and analysis of period degenerations.
result Some polynomials yield globally minimizing geodesics, with conjectures on the independence of cut time.
Study of degenerating maps to Riemannian manifolds, proving asymptotic limits and existence of minimal cylinders.
problem Blow-up analysis and behavior of maps from degenerating surfaces to compact manifolds.
method Blow-up analysis, generalized energy identities, neck asymptotic limits, evolution system for minimal cylinders.
result Asymptotic limits of necks are geodesics or geodesic-like curves, confirming conjectures.
The paper studies hanging chains and surfaces in degenerate geometries.
problem Investigating hanging chains and surfaces in simply isotropic plane and space.
method Characterizing catenaries and proving them as minimal surfaces in the simply isotropic space.
result The simply isotropic catenary is the generating curve of a minimal surface of revolution.
Study Lie algebras with complex structures, focusing on degenerations and deformations.
problem Understanding the space of Lie algebras with complex structures and their transformations.
method Identifying invariants that remain consistent under degenerations and applying to four-dimensional case.
result Found invariants that help in understanding the behavior of Lie algebras under complex structures.
Minimal 7D hypersurfaces degenerate under stability or bounded index constraints.
problem Degeneration of minimal hypersurfaces under stability or bounded index constraints.
method Analysis of sequences of minimal hypersurfaces, parameterization with controlled maps, and topological finiteness results.
result Minimal hypersurfaces can degenerate to singular ones with controlled geometry, topology, and singular set.
We study one parameter degenerations of complex projective manifolds by introducing certain type of Hodge metrics coming from the pluricanonical forms. We show that degenerations with at most canonical singularities are all in the finite distance boundary of moduli spaces. We also propose the converse to be true in the…
Minimal submanifolds are found as energy concentration sets in variational problems.
problem Understanding the structure of minimal submanifolds in codimension two.
method Purely variational approach, extending previous work on geodesics.
result Non-degenerate minimal submanifolds can be derived from critical maps of the Ginzburg-Landau functional.
Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.
problem Estimating exceptional values and classifying degenerate surfaces in Minkowski spacetime.
method Generalizing Fujimoto's theorem, estimating upper bounds, introducing conjugate similarity, and establishing structure theorems.
result Sharp contrast to Bernstein type results for minimal surfaces, estimating upper bounds of exceptional values.
New degenerate free boundary minimal annuli found in spherical caps, challenging uniqueness.
problem Non-uniqueness in spherical caps beyond the hemisphere.
method Analyzing a family of embedded free boundary minimal annuli in geodesic balls.
result Degenerate annuli exist, contradicting the Naff-Zhu uniqueness hypothesis.
The paper proves strict convexity of the Mabuchi functional for geodesics connecting energy minimizers.
problem Proving strict convexity of the Mabuchi functional for geodesics.
method Explicit formula for the complex Hessian of the weighted log-Bergman kernel, and proof by showing geodesics must be non-degenerate and smooth.
result Strict convexity of the Mabuchi functional along geodesics connecting energy minimizers.
The paper discovers new ways Riemann surfaces can degenerate.
problem Understanding degeneration of infinite-type Riemann surfaces.
method Constructing a concrete example to prove degeneration phenomena.
result Existence of degenerations in Bers boundary of infinite-type surfaces.
Paper shows examples of hyperbolic cone structures degenerating with decreasing cone angles.
problem Degeneration of hyperbolic cone structures with specific cone angles.
method Constructed examples of hyperbolic cone structures on a certain alternating link in the thickened torus.
result Example of degeneration of hyperbolic cone structures with decreasing cone angles less than 2π.
Study on Calabi-Yau metrics collapsing and complex structure degenerations.
problem Understanding the behavior of Calabi-Yau metrics on degenerating families of manifolds.
method Gluing and singular perturbation techniques, construction of Kähler metrics with torus symmetry.
result Explicit and precise relationships between metric collapsing and complex structure degenerations established in all dimensions.
Develops new Poisson structures for moduli spaces.
problem Creating Poisson structures for moduli spaces.
method Introduces quasi Poisson and quasi Hamiltonian structures, novel momentum mappings.
result Bijective correspondence between quasi Poisson and quasi Hamiltonian structures.
An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping γ: X ---> G (n, N) satisfies 0 < rank γ< n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the foca…
The paper proves smoothness of almost-minimizers' boundaries near the free boundary.
problem Minimizing degenerate area functionals with weighted boundary conditions.
method Epsilon-regularity theorem applied to almost-minimizers.
result Almost-minimizers' boundaries are C1,γ0-smooth, orthogonal to the boundary Ω. Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
Paper proves existence of minimal surfaces in hyperbolic 3-manifolds.
problem Existence of minimal surfaces in non-compact 3-manifolds.
method Analyzes specific cases of hyperbolic 3-manifolds with bounded geometry and rank-1 cusps.
result Proves existence of minimal surfaces in various hyperbolic 3-manifolds.
Introduces a new geometric structure for statistical manifolds with degenerate metrics.
problem Degenerate metrics in statistical manifolds affect geometric structures and applications.
method Introduces quasi-Codazzi structure for degenerate metrics and coherent tangent bundles.
result Generalizes geometric structures and relations for statistical models with degenerate metrics.
Noise causes learning plateaus in neural networks.
problem Plateau phenomena in online learning due to vanishing gradients.
method Analysis of stochastic gradient descent in multi-layer perceptrons.
result Noise induces synchronisation leading to strong plateaus.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and the…
We provide a complete list of two- and three-component Poisson structures of hydrodynamic type with degenerate metric, and study their homogeneous deformations. In the non-degenerate case any such deformation is trivial, that is, can be obtained via Miura transformation. We demonstrate that in the degenerate case this …
This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…
Extends arguments to limit structure in Calabi-Yau degenerations.
problem Understanding Gromov-Hausdorff limits in degenerating Calabi-Yau manifolds.
method Reduces conjecture to partial second-order estimate.
result Extends arguments to new settings.
Paper studies invariant distributions of bi-Hamiltonian structures.
problem Integrability of invariant distributions in bi-Hamiltonian structures.
method Description and investigation of invariant distributions.
result All invariant distributions of non-degenerate bi-Hamiltonian structures are described.
Constructs surfaces by gluing Delaunay ends to a minimal n-noid.
problem Creating constant mean curvature surfaces.
method Using the DPW method to glue n half Delaunay surfaces to a minimal n-noid.
result Constructs surfaces with constant mean curvature.