Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.
problem Proving Gromov's conjecture on total mean curvature of fill-ins.
method Surgery to reduce to fill-ins of spheres, positive mass theorems, and quantitative surgery process.
result Proves Gromov's conjecture on total mean curvature in various cases.
Study shows certain spin manifolds can't meet DEC condition.
problem Non-existence of spin fill-ins meeting DEC condition.
method Analyzes spin Riemannian manifolds and generalized mean curvature functions.
result Closed spin manifolds cannot satisfy DEC if curvature is large.
Researchers prove an inequality linking spin manifold fill-ins to hyperspherical radius.
problem Proving an inequality relating the infimum of mean curvature to hyperspherical radius for spin manifolds.
method Combining Hijazi-Montiel-Roldán inequality and Bär's recent theorem; providing an alternative proof based on Bär-Ballmann work.
result Proved inequality linking spin manifold fill-ins to hyperspherical radius.
New rigidity theorems for spin fill-ins with non-negative scalar curvature.
problem Mean curvature rigidity for spin fill-ins with non-negative scalar curvature.
method Two spinorial techniques: extending boundary spinors and comparison using index theory.
result New Witten-type integral inequality for the mass of asymptotically Schwarzschild manifolds.
Upper bound for total mean curvature of spin fill-ins is proven.
problem Bounding the total mean curvature of spin Riemannian manifolds.
method Proving Gromov's conjecture for spin manifolds with specific conditions.
result Explicit upper bounds for total mean curvature are derived under various conditions.
In this paper we study Spin(7)-instantons on asymptotically conical Spin(7)-orbifolds (and manifolds) obtained by filling in certain squashed 3-Sasakian 7-manifolds. We construct a 1-parameter family of explicit Spin(7)-instantons. Taking the parameter to infinity, the family (a) bubbles o…
Proves positive mass theorems for specific ALF and ALG manifolds.
problem Proving positive mass theorems for ALF and ALG manifolds.
method Reduces the problem to non-existence of positive scalar curvature metrics on closed manifolds.
result Shows that incompressible conditions for S1 and T2 are sufficient for nonnegativity of mass. Upper estimate of Dirac eigenvalue linked to hyperspherical radius.
problem Estimating the smallest eigenvalue of the Dirac operator.
method Proved an upper estimate of the smallest eigenvalue in terms of hyperspherical radius.
result Combining with known lower estimates, geometric consequences are derived.
Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
Constructs fill-ins with scalar curvature lower bounds for geometric applications.
problem Realizing (n−1)-dimensional manifolds as boundaries of higher-dimensional ones with controlled scalar curvature. method Variations of an argument by Miao and the author, constructing fill-ins with different scalar curvature lower bounds.
result Illustrates applications to geometric inequalities in general relativity, including mass bounds and Penrose inequalities.
Generalizes nonnegativity result for Brown-York mass using noncompact fill-ins.
problem Proving nonnegativity of Brown-York mass for noncompact fill-ins.
method Extending positive mass theorem to noncompact and shielded fill-ins.
result Proves nonnegativity of mean curvature for noncompact NNSC fill-ins.
Paper discusses quasilocal mass and fill-ins, proving positivity and exploring definitions.
problem Exploring and defining quasilocal mass and fill-ins in general relativity.
method Analyzes several proposals of quasilocal mass based on Hamiltonian formulation and proves positivity under certain conditions.
result Positivity of Wang-Yau energy under a more general condition.
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data (Σ,γ,H). We prove that given a metric γ on Sn−1 (3≤n≤7), (Sn−1,γ,H) admits no fill-in of NNSC metrics provided the prescribed mean cur…
We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown--York mass and the other is new. We argue by recasting the setup to the study of mean-convex fill-ins with nonnega…
The paper solves a problem related to scalar curvature and boundary metrics.
problem Proving the extensibility of boundary metrics to positive scalar curvature metrics.
method Introducing a fill-in invariant and proving relationships with positive mass theorems.
result The positive mass theorem for asymptotically hyperbolic manifolds implies the same for asymptotically flat manifolds.
Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown--York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of nonnegative sca…
Study bounds total mean curvature of fill-ins with scalar curvature constraints.
problem Bounding total mean curvature of fill-ins with scalar curvature constraints.
method Combines techniques from Shi-Tam, Shi-Wang-Wei, and recent work on systolic inequality.
result Sharp constant for total mean curvature estimate when boundary metric is flat.
MaskGAN fills in text blanks more flexibly and effectively.
problem Handling complex manipulations on given tokens in conditional text generation.
method Decomposed conditional text generation into make-a-blank and fill-in-the-blank tasks, and used hierarchical multi-agent reinforcement learning.
result The model produces realistic texts without compromising quality and diversity.
Paper proves flat 3-manifolds with positive mass have unique isoperimetric surfaces.
problem Finding unique isoperimetric surfaces in flat 3-manifolds.
method Used 'fill-in' argument and sharp isoperimetric inequality.
result Each leaf of the canonical foliation is the unique isoperimetric surface.
The paper proves conditions for positive scalar curvature metrics on manifolds with incompressible hypersurfaces.
problem Conditions for the existence of metrics with positive scalar curvature on manifolds with incompressible hypersurfaces.
method Analyzing surgeries and applying the positive mass theorem with incompressible conditions.
result Establishes positive mass theorem with incompressible conditions for specific manifolds.
In this paper, we extend the T-duality isomorphism by Gualtieri and Cavalcanti, from invariant exact Courant algebroids, to exotic exact Courant algebroids such that the momentum and winding numbers are exchanged, filling in a gap in the literature.
In this paper, we present a new approach to the construction of Einstein metrics by a generalization of Thurston's Dehn filling. In particular in dimension 3, we will obtain an analytic proof of Thurston's result.
Latent variable models can be used to probabilistically "fill-in" missing data entries. The variational autoencoder architecture (Kingma and Welling, 2014; Rezende et al., 2014) includes a "recognition" or "encoder" network that infers the latent variables given the data variables. However, it is not clear how to handl…
In this paper we show that for a Berger metric g^ on S3, the non-positively curved conformally compact Einstein metric on the 4-ball B1(0) with (S3,[g^]) as its conformal infinity is unique up to isometries and it is the metric constructed by Pedersen \cite{Pedersen}. In particular, since in \ci…
Let Sn be a punctured Riemann spheres S2\{x1,...,xn}. In this paper, we investigate pseudo-Anosov maps on Sn that are isotopic to the identity on Sn∪{xn} and have the smallest possible dilatations. We show that those maps cannot be obtained from Thurston's construction (that i…
MaskCycleGAN-VC improves voice conversion without parallel data.
problem Limited ability to convert mel-spectrogram data without parallel data.
method Integrates a novel auxiliary task called filling in frames (FIF) to learn time-frequency structures.
result MaskCycleGAN-VC outperforms existing methods with similar model size.
The compact exceptional Lie groups F4, E6, E7 and E8 have spinor groups as a subgroup as follows: E8 \supset Ss(16) \supset Spin(15) \supset Spin(14) \supset Spin(13), E7 \supset Spin(12) \supset Spin(11), E6 \supset Spin(10), F4 \supset Spin(9) \supset Spin(8) \supset Spin(7) \supset \cdot \cdot \cdot \supset Spin(1) …
A closed totally geodesic surface in the figure eight knot complement remains incompressible in all but finitely many Dehn fillings. In this paper, we show that there is no universal upper bound on the number of such fillings, independent of the surface. This answers a question of Ying-Qing Wu.
This paper improves VAE-based imputation of FX implied volatilities, reducing errors and handling uncertainty.
problem Imputing missing implied volatilities for FX options.
method Modified VAE architecture and handling uncertainty.
result Significant performance improvements, nearly halving error in low missingness regimes.
The paper explores spin^h structures and their obstructions.
problem Understanding which manifolds are spin^h.
method Analyzing the fifth Stiefel-Whitney class and constructing generalised spin structures.
result Every compact orientable manifold of dimension 7 or lower is spin^h, and there are higher-dimensional manifolds that are not.
Study mSpin(7)-dDT connections on manifolds with mSpin(7)-structures.
problem Understanding moduli spaces of mSpin(7)-dDT connections. method Introduced and studied mSpin(7)-dDT connections using fully nonlinear PDEs. result Moduli space MmSpin(7)′ has finite expected dimension and smoothness under certain conditions. New Spin(7)-instantons constructed on Joyce's manifold.
problem Constructing Spin(7)-instantons on Joyce's compact manifold. method Gluing non-flat connections on local model spaces to a flat connection on the Spin(7)-orbifold. result More than 20,000 new four-parameter families of Spin(7)-instantons. This is an introduction to the construction of higher-dimensional knots by spinning methods. Simple spinning of classical knots was introduced by E. Artin in 1926, and several generalizations have followed. These include twist spinning, superspinning or p-spinning, frame spinning, roll spinning, and deform spinning. We…
Study finds conditions for minimal surfaces in noncompact spaces.
problem Existence of minimal surfaces in noncompact spaces with nonnegative scalar curvature.
method Conditions on a 2-sphere and its boundary.
result Existence of a closed minimal surface homologous to the boundary.
In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).
New mSpinh-manifolds studied for their properties.
problem Understanding new manifold classifications.
method Exploring mSpinh-manifolds as a new category. result Highlights of mSpinh-manifolds discussed. Study on harmonic flow of Spin(7)-structures in 8D manifolds.
problem Comparing isometric Spin(7)-structures.
method Developed a notion of harmonicity for Spin(7)-structures and studied the harmonic flow.
result Analytical properties of the harmonic flow of Spin(7)-structures presented.
Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.
problem Understanding higher spin Killing spinors on 3D manifolds.
method Definition and detailed study of higher spin Killing spinors in arbitrary dimension, focusing on 3D manifolds. Rigidity result and explicit expressions for 3-sphere and 3-hyperbolic space.
result Proved a rigidity result for 3D manifolds admitting higher spin Killing spinors and provided explicit expressions for these spinors.
It is well known that certain combinations of configuration space integrals defined by Bott and Taubes produce cohomology classes of spaces of knots. The literature surrounding this important fact, however, is somewhat incomplete and lacking in detail. The aim of this paper is to fill in the gaps as well as summarize t…
The paper characterizes new invariant spinr spinors on projective spaces.
problem Characterizing new invariant spinr spinors on projective spaces. method Adapting spin representation via exterior forms to the generalised spinr context. result Complete description of the space of invariant spinr spinors for CPn, HPn, and OP2. Study metrics with specific spectral properties to compute Bartnik and Bartnik-Bray masses efficiently.
problem Compute Bartnik and Bartnik-Bray masses efficiently for metrics with specific spectral properties.
method Spectral generalization of positive scalar curvature, applying Codá Marques's path-connectedness theorem, and efficient constructions for scalar-nonnegative fill-in problem.
result Compute Bartnik and Bartnik-Bray masses efficiently for metrics with -Δ + kR ≥ 0.
We show that the pseudoconcave holes of some naturally arising class of manifolds, called hyperconcave ends, can be filled in, including the case of complex dimension 2 . As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general tha…
We explore differential and algebraic operations on the exterior product of spinor representations and their twists that give rise to cohomology, the spin cohomology. A linear differential operator d is introduced which is associated to a connection ∇ and a parallel spinor ζ, ∇ζ=0, and the algebraic o…
We address the problem of surface inpainting, which aims to fill in holes or missing regions on a Riemann surface based on its surface geometry. In practical situation, surfaces obtained from range scanners often have holes where the 3D models are incomplete. In order to analyze the 3D shapes effectively, restoring the…
Proves almost flat spin^c manifolds bound compact manifolds.
problem Proving almost flat spin^c manifolds bound compact manifolds.
method Long-standing conjecture of Farrell--Zdravkovska and S. T. Yau settled.
result Every almost flat spin^c manifold bounds a compact orientable manifold.
Rigidity of elliptic genera proven for non-spin manifolds with S1-action.
problem Rigidity of elliptic genera for non-spin manifolds with S1-action. method Analysis of universal covering spin condition and π2(M) for rigidity. result Rigidity of elliptic genera is proven for spin universal coverings but not for non-spin universal coverings.