arXiv research

Compact curve solution emerges from non-compact curve.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as $t \to -\infty$, to two self-similar complete non-compact solutions to the Yamabe flow moving in opposite directions. They are type I ancient solutions.

Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.

Study finds obstacles to solutions for specific equations on compact surfaces.

Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.

Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

Compact, non-convex curve flows are created.

Survey on Kähler-Ricci flow solutions.

Complete solutions found for Toda equations on non-compact surfaces.

In this paper, we give a complete classification of $κ$-solutions of Kähaler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.

Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.

The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.

Two ancient solutions to Gauss curvature flow are identified for cylinders.

Study axisymmetric $σ_k$-Nirenberg problem on spheres.

Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (-)-connection} on 6-dimensional non-Kaehler nilmanifold are presented. In the conformally compact case the dilaton is determined by t…

Study on compactness and blow-up of solutions for Yamabe problems on manifolds with non-umbilic boundaries.

We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.

Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.

We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.

We consider an ancient solution $g(\cdot,t)$ of the Ricci flow on a compact surface that exists for $t\in (-\infty,T)$ and becomes spherical at time $t=T$. We prove that the metric $g(\cdot,t)$ is either a family of contracting spheres, which is a type I ancient solution, or a Rosenau solution, which is a type II ancie…

We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which we describe. This analysis applies in particular to the ancient solution constru…

Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.

We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $Σ$ of R^d, with $d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a re…

Defines and studies solutions to complex equations on Hermitian manifolds.

Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.

Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.

Study ancient solutions on noncompact steady Ricci solitons, proving types of ancient solutions.

Study on transverse Ricci solitons on compact foliated manifolds.

The paper resolves compactness and non-compactness for fourth- and sixth-order Q-curvature problems.

We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as $t\to-\infty$ and we give their precise asymptotic description. This description applies …

The paper proves estimates for a specific flow on compact manifolds.

The paper finds sign-changing solutions for a specific type of elliptic equation.

Paper establishes estimates for solutions on compact manifolds.

Study on blow-up behavior of sign-changing solutions for Yamabe equation.

For a sequence of blow up solutions of the Yamabe equation on non-locally confonformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points…

We study the evolution of complete non-compact convex hypersurfaces in $\mathbb{R}^{n+1}$ by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proo…

We generalize the ancient solutions of the Ricci flow on certain principal ${\rm SO}(3)$ bundles over compact quaternionic Kähler manifolds constructed by Bakas, Kong, and Ni to certain $RP^3$ fibre bundles over a product of two compact quaternionic Kähler manifolds. The ancient solutions are of Type I, $κ$-noncollapse…

New ancient solutions found for curvature flow in 2D.

The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.

We construct explicit compact solutions with non-zero field strength, non-flat instanton and constant dilaton to the heterotic string equations in dimensions seven and eight. We present a quadratic condition on the curvature which is necessary and sufficient the heterotic supersymmetry and the anomaly cancellation to i…

In this note, we prove the existence of weak solutions of the Chern-Ricci flow through blow downs of exceptional curves, as well as backwards smooth convergence away from the exceptional curves on compact complex surfaces. The smoothing property for the Chern-Ricci flow is also obtained on compact Hermitian manifolds o…

The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

We construct a compact, convex ancient solution of mean curvature flow in $\mathbb R^{n+1}$ with $O(1)\times O(n)$ symmetry that lies in a slab of width $π$. We provide detailed asymptotics for this solution and show that, up to rigid motions, it is the only compact, convex, $O(n)$-invariant ancient solution that lies …

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stabilit…

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$ topology as $t\rightarrow\infty$. Up to scaling, the limit function is a solution of t…