New ancient compact solutions found for Yamabe flow.
problem Finding compact solutions to the Yamabe flow.
method Constructed rotationally symmetric ancient compact solutions.
result Found type I ancient compact solutions converging to self-similar solutions.
Compact curve solution emerges from non-compact curve.
problem Constructing solutions from non-compact curves.
method Slingshot solution to curve shortening flow.
result Compact embedded solution exists for a finite time.
The paper classifies κ-solutions of Kähler-Ricci flow on compact manifolds.
problem Classifying κ-solutions of Kähler-Ricci flow on compact complex manifolds.
method Complete classification through quotients of products of irreducible compact Hermitian symmetric manifolds.
result κ-solutions of Kähler-Ricci flow on compact manifolds must be quotients of products of irreducible compact Hermitian symmetric manifolds.
We construct new ancient compact solutions to the Yamabe flow. Our solutions are rotationally symmetric and converge, as t→−∞, 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.
problem Existence and uniqueness of viscosity solutions to complex Hessian equations.
method Proves existence and uniqueness using viscosity solutions and determinant domination conditions.
result Viscosity solutions exist and are unique under certain conditions.
Study finds obstacles to solutions for specific equations on compact surfaces.
problem Existence of solutions to self-dual equations on compact surfaces.
method Depends on Higgs field zeroes and vortex number.
result Infinitely many Higgs fields for which solutions cannot exist.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
Stability and Hölder continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Establishing Hölder continuity of solutions to complex Monge-Ampère equations.
method Stability result for solutions in Lp space, Hölder continuity proof. result Solutions are Hölder continuous with the same exponent as in the Kähler case.
Solves long-term solutions for a specific equation on compact manifolds.
problem Existence and uniqueness of solutions to the parabolic Monge-Ampère equation on compact almost Hermitian manifolds.
method Proves long-term existence and uniqueness using scaling and convergence arguments.
result Normalization of solutions converges to a smooth function in C∞ topology as tightarrow∞. Compact, non-convex curve flows are created.
problem Creating compact, non-convex ancient solutions for curve shortening flow.
method Constructed an ancient solution asymptotic to Yin-Yang curve.
result Compact, non-convex ancient solutions for curve shortening flow are demonstrated.
Survey on Kähler-Ricci flow solutions.
problem Existence of solutions for Kähler-Ricci flow.
method Survey of recent developments.
result Discussion of solutions existing for all positive times.
Weak solutions found for Chern-Ricci flow on complex surfaces.
problem Existence of weak solutions for Chern-Ricci flow on compact complex surfaces.
method Through blow downs of exceptional curves and backwards smooth convergence.
result Existence of weak solutions and smoothing property proved.
Complete solutions found for Toda equations on non-compact surfaces.
problem Solving Toda equations on non-compact Riemann surfaces.
method Introduced complete solutions and proved existence and uniqueness using Toda equations and harmonic bundle techniques.
result Existence and uniqueness of complete solutions to Toda equations on non-compact Riemann surfaces.
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
The study finds continuous solutions to complex Hessian equations on compact Hermitian manifolds.
problem Finding continuous solutions to complex Hessian equations on compact Hermitian manifolds.
method Deriving an L∞-estimate for bounded solutions to the complex m-th Hessian equations on compact Hermitian manifolds, assuming a positive right-hand side in the Orlicz space Lmn(logL)n(h∘log∘logL)n. result Establishing the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.
Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.
problem Gradient obstacle problems on compact Riemannian manifolds.
method Uniform semiconcavity estimates and fine convergence results for solutions and free boundaries.
result The elastic and λ-elastic sets of solutions converge to the cut locus and λ-cut locus of the manifold. We study compactness of solutions to the Yamabe problem on Riemannian manifolds which are not locally conformally flat.
Two ancient solutions to Gauss curvature flow are identified for cylinders.
problem Classifying ancient solutions to Gauss curvature flow in cylinders.
method Assumption of cylinder cross-section bounded convexity, analysis of asymptotic behavior.
result Only two ancient solutions identified: translating soliton and compact oval solution.
Study axisymmetric σk-Nirenberg problem on spheres.
problem Prescribing σk-curvature for axisymmetric metrics on spheres. method Compactness, non-compactness, existence, and non-existence results proved based on curvature function behaviors.
result Existence and non-existence of solutions depend on curvature function behaviors near poles.
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.
problem Compactness and blow-up behavior of solutions to the Yamabe boundary problem on manifolds with non-umbilic boundaries.
method Analysis of stability and blow-up sequences for solutions under perturbations of mean curvature and scalar curvature.
result Existence of a blowing-up sequence of solutions when perturbing the mean curvature from above or below with a function having a large positive maximum.
In 2D, a unique solution is proven for a specific equation.
problem Uniqueness of solutions to a specific equation in 2D.
method Analysis on compact Riemannian surfaces without boundary.
result Uniqueness of solutions proven in 2D.
Compact mean curvature flow solutions with bounded curvature in high dimensions are constructed.
problem Constructing compact mean curvature flow solutions with bounded mean curvature.
method Following Velázquez, Guo, Sesum, and Stolarski's arguments, constructing solutions in \(\mathbb{R}^n\) with \(n \geq 8\).
result Compact mean curvature flow solutions with bounded mean curvature in \(\mathbb{R}^n\) are constructed.
We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.
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.
Paper proves existence of solutions for a specific differential equation on compact manifolds.
problem Existence of solutions for a non-linear differential equation on compact Riemannian manifolds.
method Lower and upper solutions method.
result Established existence of a smooth positive solution for the equation.
We consider an ancient solution g(⋅,t) of the Ricci flow on a compact surface that exists for t∈(−∞,T) and becomes spherical at time t=T. We prove that the metric g(⋅,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…
Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Stability and continuity of solutions to degenerate complex Monge-Ampère equations.
method Analysis of Hölder continuity and global continuity of solutions.
result Established uniform diameter bound for the twisted Chern-Ricci flow.
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≥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…
Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.
problem Existence and asymptotic behavior of solutions to Navier-Stokes equations on non-compact manifolds.
method Used Lp−Lq-dispersive and smoothing estimates of the Stokes semigroup, fixed point arguments, and Gronwall's inequality. result Established existence and exponential decay of almost periodic and asymptotically almost periodic mild solutions.
Defines and studies solutions to complex equations on Hermitian manifolds.
problem Solving complex equations on Hermitian manifolds.
method Extending recent theories, defines and studies pluripotential solutions to degenerate parabolic complex Monge-Ampère equations.
result Establishes existence and uniqueness of weak Chern-Ricci flow on complex compact varieties with log terminal singularities.
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces proved.
problem Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces.
method Analyzing bounded solutions on reduced, locally irreducible compact Kähler spaces.
result Proves continuity of solutions, affirming conjectures and solving open problems.
Study ancient Ricci flow solutions, proving unique asymptotic behavior.
problem Understanding unique asymptotics of compact ancient solutions to 3D Ricci flow.
method Analyzing noncollapsed compact ancient solutions, proving asymptotic behavior.
result Proves unique asymptotic behavior for compact ancient solutions.
Study ancient solutions on noncompact steady Ricci solitons, proving types of ancient solutions.
problem Classify ancient solutions on noncompact steady gradient Ricci solitons.
method Apply Perelman's L-geodesic theory to analyze blow-down solutions. result Prove that compact split ancient solutions are of type I.
Study on transverse Ricci solitons on compact foliated manifolds.
problem Characterizing transverse Ricci solitons on compact foliated manifolds.
method Investigation of self-similar solutions of the transverse Ricci flow, analysis of taut Riemannian foliations.
result Established relations between taut Riemannian foliations and transverse Ricci solitons, found examples of transverse Ricci solitons.
The paper resolves compactness and non-compactness for fourth- and sixth-order Q-curvature problems.
problem Compactness and non-compactness of fourth- and sixth-order Q-curvature problems.
method Transformed linearized equations into overdetermined systems revealing algebraic structures.
result Proves compactness for fourth-order Q-curvature problems in dimensions 5 to 24, sixth-order in 7 to 26.
Study finds unique asymptotic behavior for certain 3D Ricci flow solutions.
problem Understanding the long-term behavior of specific 3D Ricci flow solutions.
method Analyzes rotationally and reflection symmetric ancient solutions.
result Ancient solutions are either spheres or have unique asymptotic behavior.
The paper proves estimates for a specific flow on compact manifolds.
problem Proving estimates for the Ricci-Bourguignon flow.
method Hamilton-Ivey estimates for the Ricci-Bourguignon flow on compact manifolds with n=3 and ρ<0. result Compact ancient solutions have nonnegative sectional curvature for all negative ρ. The paper finds sign-changing solutions for a specific type of elliptic equation.
problem Existence of sign-changing solutions for a Yamabe type equation.
method Investigates a critical elliptic equation with a Yamabe type operator on a compact manifold with boundary.
result Existence of sign-changing solutions assured under certain geometric conditions.
Study on blow-up behavior of sign-changing solutions for Yamabe equation.
problem Blow-up behavior of sign-changing solutions for Yamabe equation.
method Construction of a smooth metric on space forms to prove blow-up at lowest energy level.
result Blow-up occurs at the lowest energy level for sign-changing solutions in dimensions 11 to 24.
Paper establishes estimates for solutions on compact manifolds.
problem Solving fully non-linear equations on compact almost Hermitian manifolds.
method Establishes a priori estimates for solutions.
result Solves complex Hessian and Monge-Ampère equations.
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 find nontrivial solutions to a Ginzburg-Landau equation on compact manifolds.
problem Finding nontrivial solutions to a specific Ginzburg-Landau equation on compact manifolds.
method Using min-max techniques to construct solutions whose energy grows logarithmically with a small parameter.
result The energy of constructed solutions concentrates on a nontrivial stationary, rectifiable (n−2)-varifold. Ancient solution found in 3D space with specific symmetry properties.
problem Finding ancient solutions with specific symmetry and geometric constraints in 3D space.
method Constructed a compact, convex ancient solution with O(1)imesO(n) symmetry in a slab of width π. result The only compact, convex, O(n)-invariant ancient solution in a slab of width π. New ancient solutions found for curvature flow in 2D.
problem Ancient solutions for curvature flow in 2D.
method Constructing and classifying convex ancient solutions.
result All convex ancient solutions classified for α∈(32,1). 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…
The paper constructs ancient solutions to curvature flows in bounded and unbounded regions.
problem Understanding ancient solutions to curvature flows in bounded and unbounded regions.
method Constructing pancake-like and sausage-like ancient compact solutions.
result Ancient solutions to curvature flows in bounded and unbounded regions.
The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.
problem Uniqueness and symmetry of self-similar solutions in warped product spaces.
method Analysis of curvature flows with homogeneous speed functions in warped product spaces.
result Compact star-shaped self-similar solutions in warped product spaces are slices.