arXiv research

Paper proves uniqueness of special Lagrangian pair in Calabi-Yau 3-fold.

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

Unique Poincaré type cscK metric with singularity at smooth divisor is unique up to holomorphic transformations.

Unique solution found for Demailly's equation on stable bundles.

The paper solves problems related to curvature on a 3-sphere.

We prove Thurston's bending measure conjecture for quasifuchsian once punctured torus groups. The conjecture states that the bending measures of the two components of the convex hull boundary uniquely determine the group.

Reformulated Markov's conjecture in combinatorial terms.

We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…

The Clifford torus is unique when its isoperimetric ratio is prescribed.

Unified proof of Aigner's conjectures using geodesics.

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kahler structure of complex projective space, and of Yau's resolution of the Severi Conjecture.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

Study biharmonic hypersurfaces in spheres, proving unique continuation theorem.

We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies…

Bonahon conjectured that compact convex cores with totally geodesic boundary uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This paper proves Bonahon's conjecture. The proofs extend the techniques of Besson-Courtois-Gallot.

A minimal hypersurface in a sphere is uniquely determined.

A smooth diffeomorphism is said to be distributionally uniquely ergodic (DUE for short) when it is uniquely ergodic and its unique invariant probability measure is the only invariant distribution (up to multiplication by a constant). Ergodic translations on tori are classical examples of DUE diffeomorphisms. In this ar…

Curve shortening flow is not unique on certain metrics.

Sum of Lagrange numbers equals a specific formula.

This paper has been withdrawn by author due to an error in the proof.

We prove uniqueness of solutions to complex Monge-Ampère equations for small temperature.

The paper confirms conjectures about ancient ovals and provides counterexamples.

Gronwall conjecture states that a planar 3-web which admits more than one distinct linearization is locally equivalent to an algebraic web. We give a partial answer to the conjecture in the affirmative for the class of planar 3-webs with the web curvature that vanishes to order three at a point. The differential relati…

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a,b,c), where a, b, and c are in increasing order, holds whenever $c$ is a prime power.

Proves uniqueness of certain $S^1$-symmetric gravitational instantons.

Solves a generalized Monge-Ampère equation on Kähler surfaces, proving a conjecture.

Proves uniqueness of translators in 3D space.

We proved two Three Circles Theorems for harmonic functions on manifolds in integral sense. As one application, on manifold with nonnegative Ricci curvature, whose tangent cone at infinity is the unique metric cone with unique conic measure, we showed the existence of nonconstant harmonic functions with polynomial grow…

Solves the quintuple bubble problem on spheres and Euclidean spaces.

Researchers prove a complex geometric conjecture about certain manifolds.

The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fa…

Thurston's Ending Lamination Conjecture states that a hyperbolic 3-manifold N with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups; the general case when N has incompressible ends relative to i…

We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebra…

We show that Lawson's bipolar surface $\tildeτ_{3,1}$ is after stereographic projection the unique minimizer among immersed Klein bottles in its conformal class. We conjecture that it actually is the unique minimizer among immersed Klein bottles into $\mathbb{R}^n$, $n\geq 4$, whose existence the authors and P. Breunin…

We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.

Either fibered knots supporting the tight contact structure are unique in their smooth concordance class or there exists a fibered counterexample to the Slice-Ribbon Conjecture.

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

Proves Arnold-Thom conjecture for surfaces' arrival times.

We study the curvature condition which uniquely characterizes the hemisphere. In particular, we prove the Min-Oo conjecture for hypersurfaces in Euclidean space and hyperbolic space.

Gravitational instantons are non-Kähler, providing a counterexample to Euclidean Black Hole Uniqueness.

Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.

Proves uniqueness of holomorphic quilts on surfaces.

New 1-parameter family of ovals identified in 4d Ricci flow classification.

We establish a black hole uniqueness theorem for Schwarzschild-de Sitter spacetime, also called Kottler spacetime, which satisfies Einstein's field equations of general relativity with positive cosmological constant. Our result concerns the class of static vacuum spacetimes with compact spacelike slices and regular max…

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bund…

Proves conjecture simplifying mapping class group action on Steinberg module.

Unique continuation property for measures in high dimensions.