Proves minimality of tensor varieties, generalizing previous results.
problem Finding minimality conditions for tensor varieties.
method Using Lawlor's curvature criterion and deriving Lawlor's ODE.
result Proves minimality of a class of tensor varieties except for one case.
This paper proves area-minimizing cones over Grassmannian manifolds.
problem Determine if cones over Grassmannian manifolds are area-minimizing.
method Detailed descriptions of embedding maps using Hermitian orthogonal projectors, re-proving area-minimization using Lawlor's Curvature Criterion.
result All cones over Grassmannian manifolds are area-minimizing except for oriented real Grassmannians.
Constructing translating solitons from Lagrangian Grim Reapers.
problem Creating Lagrangian translating solitons from intersections of Grim Reapers.
method Desingularizing intersections with special Lagrangian Lawlor necks.
result Constructing Lagrangian translating solitons with multiple ends and loops.
Study on singularities in Lagrangian mean curvature flow with special Lagrangian cones.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analysis of tangent flows and blowup limits of special Lagrangian cones.
result Uniqueness of tangent flows in dimension two and any dimension when the link is connected.
Paper detects duality obstruction in smooth calibrations.
problem Detecting duality obstruction in smooth calibrations.
method Examine Lawlor cones and calibrations, use gluing results.
result Existence of Lawlor cones without smooth calibrations.
The paper constructs solutions with infinite-time singularities in Lagrangian mean curvature flow.
problem Infinite-time singularities in Lagrangian mean curvature flow.
method Constructing solutions by gluing special Lagrangian 'Lawlor necks' and analyzing dynamics of neck size.
result The flow decomposes initial data into a union of special Lagrangians intersecting at one point.
Explains examples of Lagrangian flow with circle symmetry.
problem Understanding Lagrangian flow with symmetry.
method Examining specific examples in C2. result Shows various types of flow, including compact and non-compact.
Flow preserves Lagrangian condition in Calabi-Yau manifolds.
problem Preserving Lagrangian condition in Calabi-Yau manifolds with boundary.
method Introduced mixed Dirichlet-Neumann boundary condition for Lagrangian mean curvature flow.
result Proved preservation of Lagrangian condition under flow.
We prove two main results: (a) Suppose L is a closed, embedded, exact special Lagrangian m-fold in Cm for m≥3 asymptotic at infinity to the union Π1∪Π2 of two transverse special Lagrangian planes Π1,Π2 in Cm. Then L is one of the explicit 'Lawlor neck' family of examples …
We study almost-calibrated, O(n)-equivariant Lagrangian mean curvature flow in Cn, and prove structural theorems about the Type I and Type II blowups of finite-time singularities. In particular, we prove that any Type I blowup of such a flow must be a special Lagrangian pair of transversely intersecting p…
We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…
We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.
Researchers create non-degenerate harmonic functions on n-dimensional space.
problem Creating non-degenerate Z2-harmonic functions on Rn. method Using a variant of ellipsoidal coordinates, the construction is explicit and involves Lawlor's necks in Cn. result First known family of non-degenerate Z2-harmonic 1-forms with compact branching sets. We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].
We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at th…
The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.
problem Stability and minimizing properties of higher codimensional surfaces in Euclidean space.
method Analyzes surfaces associated with the weighted area-functional and proves stability and minimization properties under specific conditions.
result Minimal cones with globally flat normal bundles are f-stable, and highly singular determinantal varieties and Pfaffian varieties are f-minimizing. Butscher, D. Lee, Y. Lee, and Joyce constructed a special Lagrangian submanifold by gluing a Lawlor neck into a transverse intersection point of two special Lagrangian submanifolds. We prove a uniqueness theorem for the gluing of flat special Lagrangian tori of real dimension 3 in a flat complex torus of complex dimens…
In this paper we prove the existence of families of n-dimensional complete embedded minimal submanifolds of C^n with a prescribed configuration of k>1 asymptotic planes. These submanifolds are obtained by desingularizing the intersection of the asymptotes, using a gluing theorem applied to a generalization of a special…
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
The study provides a criterion for diffeomorphism via long-time Ricci flow.
problem Understanding conditions for diffeomorphism in geometric flows.
method Long-time Ricci flow criterion for diffeomorphism.
result Affirmative answer to manifold diffeomorphism in dimension 4.
This paper proves all Pfaffian varieties are area-minimizing except hypersurfaces.
problem Proving area-minimizing property of Pfaffian varieties.
method Analyzing families of minimal real matrix varieties and proving area-minimizing property.
result All Pfaffian varieties are area-minimizing except hypersurfaces.
In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. It is an interesting open problem, also due to Manning, to determine to what extent such behaviour persists for higher dimensional manifolds. In …
We shall show that q-semipositivity of the vector bundle E over a Kähler total space X implies the Griffiths-semipositivity of the q-th direct image of O(KX/B⊗E). As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on t…
Study minimizers in large volume isoperimetric problems with a new flatness criterion.
problem Minimizers in isoperimetric problems with a compact obstacle.
method Study Plateau-type problem with free boundary, develop mesoscale flatness criterion.
result Identify isoperimetric residue in energy expansion for large volume.
Study on stable minimal hypersurfaces under Ricci curvature constraints.
problem Stability of weighted minimal hypersurfaces under Ricci curvature bounds.
method Derive geometric consequences and prove a Schoen-Yau type criterion.
result Structure theorem for three-dimensional weighted manifolds of non-negative Ricci curvature.
This paper proves a Nakai-Moishezon criterion for complex Hessian equations.
problem The solvability of complex Hessian equations on Kähler manifolds.
method Establishing a Nakai-Moishezon criterion for Kähler classes on analytic Kähler varieties.
result Proves Lejmi-Szekelyhidi's conjecture for the J-equation. Paper shows triviality criterion for certain Ricci solitons.
problem Identifying trivial non-steady gradient Ricci solitons.
method Analyzes scalar curvature and its relation to triviality.
result Triviality of solitons depends on scalar curvature equality.
Nearly spherical, positively curved surfaces are mapped from a sphere.
problem Mapping nearly spherical, positively curved surfaces from a sphere.
method Combines Ricci flow, Kim-Milman construction, and Bakry-Émery criterion.
result Every nearly spherical, positively curved surface is the contractive image of a round sphere.
Study finds criteria for surfaces with specific curvature properties.
problem Understanding Kählerian or projective structures on surfaces with non-positive curvature.
method Established a criterion for compact Hermitian surfaces with non-positive second Chern-Ricci curvature.
result Found conditions for Kählerian or projective structures on surfaces with non-positive curvature.
We introduce a CR-invariant class of Lorentzian metrics on a circle bundle over a 3-dimensional CR-structure, which we call quasi-Fefferman metrics. These metrics generalise the Fefferman metric but allow for more control of the Ricci curvature. Our main result is a criterion for embaddability of 3-dimensional CR-struc…
Analytic completeness criterion applied to constant mean curvature surfaces.
problem Determining the analytic completeness of constant mean curvature surfaces.
method Defining arc-properness and applying it to surfaces in de Sitter 3-space.
result A criterion for the analytic completeness of G-catenoids and their extensions.
In this article we prove a generalization of Weyl's criterion for the spectrum of a self-adjoint nonnegative operator on a Hilbert space. We will apply this new criterion in combination with Cheeger-Fukaya-Gromov and Cheeger-Colding theory to study the k-form essential spectrum over a complete manifold with vanishing…
Improving a result of Eschenburg and Kim we give a criterion for semisimplicity of pseudo-Riemannian extrinsic symmetric spaces in terms of the shape operator with respect to the mean curvature vector.
We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of C∞ norms on R3 admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…
Proves Chen-Lin conjecture for sphere scalar curvature problem.
problem Existence of solutions for prescribed scalar curvature on spheres.
method Criterion proving existence based on Chen-Lin conjecture.
result Proves conjecture for standard n-dimensional sphere.
Study on minimizing singular capillary cones with stability and instability results.
problem Minimizing singular capillary cones with free boundary.
method Stability criterion à la Jerison-Savin, Simons-type inequality for convex, homogeneous, symmetric functions of principal curvatures, boundary condition specific to capillary setting.
result Minimizing cones with non-sign-changing mean curvature are flat in dimensions up to 4, and non-trivial axially symmetric cones are unstable in dimensions up to 6.
Alternative solvability criterion for minimal surface equations and mean curvature flow.
problem Solvability of Dirichlet problem for minimal surface equation in non-mean convex domains.
method Introduces a structural condition from a second-order ODE to construct boundary barriers, applicable to unbounded domains and Hadamard manifolds.
result Allows solvability under geometric hypotheses different from classical Jenkins-Serrin theory, applicable to Euclidean space and mean curvature flow.
In a complete simply connected Riemannian manifold X of pinched negative curvature, we give a sharp criterion for a subset C to be the epsilon-neighbourhood of some convex subset of X, in terms of the extrinsic curvatures of the boundary of C.
The paper proves wave operator existence and completeness for Hodge Laplacians.
problem Proving the existence and completeness of wave operators for Hodge Laplacians.
method Integral criterion, probabilistic Bismut-type formulae, heat semigroup, local curvature bounds.
result Absolutely continuous spectra of Hodge Laplacians coincide under quasi-isometry.
The paper proves boundedness of a Riesz transform on weighted manifolds.
problem Establishing \(L^p\)-boundedness of the covariant Riesz transform on differential forms.
method Heat-kernel criterion, volume doubling, heat kernel estimates, curvature control, gradient bounds.
result The covariant Riesz transform is \(L^p\)-bounded for \(p>2\) on weighted Riemannian manifolds.
In this paper, we consider the problem of prescribing the scalar curvature under minimal boundary conditions on the standard four dimensional half sphere. We provide an Euler-Hopf type criterion for a given function to be a scalar curvature to a metric conformal to the standard one. Our proof involves the study of crit…
Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CPN−1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic…
Criterions for constancy of the holomorphic sectional curvature and the antiholomorphic sectional curvature are proved for almost Hermitian manifolds. It is shown, that an almost Hermitian manifold satisfying the axiom of antiholomorphic planes or the axiom of antiholomorphic spheres is a real or a complex space form.
The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.
problem Establishing convergence and long-time existence of the combinatorial p-th Calabi flow.
method Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns.
result Sharp criterion for convergence in finite case and long-time existence in infinite case for p≥2. The study establishes a criterion for the holomorphy of curvature in smooth webs and applies it to dual webs of homogeneous foliations.
problem Establishing conditions for the holomorphy of curvature in smooth webs and their duals.
method Developed an effective criterion for the holomorphy of curvature in smooth d-webs and applied it to dual webs of homogeneous foliations. result Characterized the holomorphy of the curvature of dual webs of homogeneous foliations on PC2. Study on prescribing positive curvature with conical singularities on a sphere.
problem Prescribing positive curvature with conical singularities on a sphere.
method Fine analysis of bubble trees and an area identity in the convergence process.
result Criterion for nonexistence in an open region of the prescribing data.
The paper connects curvature positivity to rational connectedness in complex geometry.
problem Establishing a geometric criterion for rational connectedness.
method Uhlenbeck-Yau's continuity method applied to mean curvature positivity.
result Holomorphic tangent bundle mean curvature positivity is equivalent to rational connectedness of compact Kähler manifolds.
The ACS criterion is verified for specific hypersurfaces in unit spheres.
problem Verifying the ACS criterion for minimal isoparametric hypersurfaces in unit spheres.
method Moment-relaxation technique and explicit extremal configurations.
result The ACS condition holds under specific conditions on principal curvatures.