arXiv research

Paper proves a rigidity result for static perfect fluids.

Proves existence and uniqueness of rotating fluid bodies in GR to second order.

Study on static perfect fluid space-time geometry and boundary estimates.

We obtain expressions for the shear and the vorticity tensors of perfect-fluid spacetimes, in terms of the divergence of the Weyl tensor. For such spacetimes, we prove that if the gradient of the energy density is parallel to the velocity, then either the expansion rate is zero, or the vorticity vanishes. This statemen…

In this paper emphasis is placed on how the behavior of the solutions of a PDE is affected by the geometry of the generalized $m$-quasi-Einstein manifold, and vice versa. Considering a $n$-dimensional generalized $m$-quasi-Einstein manifold which is conformal to a pseudo-Euclidean space, we prove the most general symme…

We prove that shear-free perfect fluid solutions of Einstein's field equations must be either expansion-free or non-rotating (as conjectured by Treciokas and Ellis) for all linear equations of state $p = w ρ$ except for six values of $w$.

New methods for constructing null fluid metrics and solving optical lift conjectures.

We found an infinite family of counterexamples to Batson's conjecture.

We describe two methods for showing that a vector can not be the f-vector of a homology d-ball. As a consequence, we disprove a conjectured characterization of the f-vectors of balls of dimension five and higher due to Billera and Lee. We also provide a construction of triangulated balls with various f-vectors. We show…

This article is a short nontechnical survey of recent progresses in fluid dynamics and differential geometry, relating a conjecture of Lars Onsager to the work of Nash on isometric embeddings.

New method uses relative capacities of geodesic balls to determine scalar curvature.

Local isoperimetric inequality holds for balls with nonpositive curvature.

Counterexamples found for knot conjectures.

In our previous article [Rad16], we investigated the asymptotic behaviour of orthogonal Bianchi class B perfect fluids close to the initial singularity and proved the Strong Cosmic Censorship conjecture in this setting. In several of the statements, the case of a stiff fluid had to be excluded. The present paper fills …

This paper proves a conjecture about unique positive harmonic functions in a ball.

Determines surgeries on chain links bounding rational homology balls using lattice-theoretic methods.

We apply Donaldson's theorem on the intersection forms of definite 4--manifolds to characterize the lens spaces which smoothly bound rational homology 4--dimensional balls. Our result implies, in particular, that every smoothly slice 2--bridge knot is ribbon, proving the ribbon conjecture for 2--bridge knots.

Fixed points of Minkowski valuations are found in specific ball neighborhoods.

New 3-manifolds bound rational 4-balls through specific operations.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

The Strong Cosmic Censorship conjecture states that for generic initial data to Einstein's field equations, the maximal globally hyperbolic development is inextendible. We prove this conjecture in the class of orthogonal Bianchi class B perfect fluids and vacuum spacetimes, by showing that unboundedness of certain curv…

The paper confirms a conjecture about convex bodies and their properties.

The paper confirms conjectures about Stein manifolds formed by quotients of the ball.

The Kneser-Poulsen conjecture says that if a finite collection of balls in a Euclidean (spherical or hyperbolic) space is rearranged so that the distance between each pair of centers does not increase, then the volume of the union of these balls does not increase as well. We give new results about central sets of subse…

The paper studies $\varphi$-static perfect fluid space-times in Einstein's General Relativity.

The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.

The minimal number of critical points is studied for smooth functions on closed manifolds.

We consider the kinematics of specific fluid spacetimes admitting timelike congruences of Ricci Solitons. These fluids includes string cloud, string fluid, perfect fluid, radially symmetric fluid, anisotropic fluid and relativistic magneto-fluid. Results are obtained and important physical aspects are discussed.

In this paper, we prove a scalar curvature rigidity result for geodesic balls in S^n. This result contrasts sharply with the recent counterexamples to Min-Oo's conjecture for the hemisphere (cf. [5]).

Spaces of polynomials are shown to be Euclidean balls.

We obtain an improved pseudolocality result for Ricci flows on two-dimensional surfaces that are initially almost-hyperbolic on large hyperbolic balls. We prove that, at the central point of the hyperbolic ball, the Gauss curvature remains close to the hyperbolic value for a time that grows exponentially in the radius …

This is the second paper of two in a series under the same title ([CRX]); both study the quantitative volume space form rigidity conjecture: a closed $n$-manifold of Ricci curvature at least $(n-1)H$, $H=\pm 1$ or $0$ is diffeomorphic to a $H$-space form if for every ball of definite size on $M$, the lifting ball on th…

Nearby pinwheels are isotopic, solving Arnold's conjecture.

This paper confirms volumes of geodesic balls can identify 4D space forms.

The Farey tree helps embed rational balls and lens spaces into complex projective space.

Study on Funk geometry volume growth and polytope flags, verifying conjectures.

Novel deep learning approach for fast, differentiable fluid simulations.

FLUID-LLM uses LLMs to predict fluid dynamics with improved accuracy.

We show that the torus knot $T_{4,9}$ bounds a smooth Möbius band in the $4$-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.

The paper divides minimal hypersurfaces in a ball into two parts.

In this paper we review the development and recent results of the Siu-Yang conjecture which is that every Kähler-Einstein compact complex manifold of complex dimension two with negative sectional curvature is biholomorphic to a compact quotient of the complex 2-ball.

We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $λ_2 - λ_1$ of Dirichlet eigenvalues, amoung domains with fixed $λ_1$. We prove an upper bound on $λ_2 - λ_1$ for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.

We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which …

The study examines perfect fluid spacetimes and their properties.

The paper finds conditions for pseudosymmetric spacetimes to be perfect fluids.

Extends Onsager's conjecture to Besov spaces on manifolds with boundary.

The study examines properties of perfect fluid spacetimes in Einstein's theory.