The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Study rigidity of geodesic balls on manifolds with boundary.
problem Rigidity of geodesic balls on manifolds with boundary.
method Combining generalized Reilly formula with Steklov-type boundary value problems to derive integral inequalities.
result Characterizations of geodesic balls in space forms.
The paper proves rigidity results for Serrin-type problems in manifolds.
problem Proving rigidity for Serrin-type problems in Riemannian manifolds.
method Integral identities and Soap Bubble theorem.
result Rigidity results for annular regions in Einstein manifolds.
Proves rigidity for maps between manifolds using degree theory and current developments.
problem Lipschitz-volume rigidity for maps between metric manifolds and Riemannian manifolds.
method Degree theory and recent developments of Lipschitz-volume rigidity for integral currents.
result Proves a Lipschitz-volume rigidity result for 1-Lipschitz maps.
Paper extends rigidity and vanishing results for totally real submanifolds under Lp-integrable conditions.
problem Rigidity and vanishing properties of totally real submanifolds in complex space forms.
method Extends results by Cuong et al. to a broader range of p-integrable conditions. result Extends the range of p for rigidity and vanishing results. Proves long-time Ricci flow existence and topological rigidity for pinched integral curvature manifolds.
problem Proving long-time existence and topological rigidity for manifolds with pinched scale-invariant integral curvature.
method Proves long-time existence of Ricci flow for manifolds with bounded curvature and pinched scale-invariant integral curvature, converging to a flat metric.
result Flow converges to a flat metric, implying topological rigidity of the manifold.
The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.
problem Finding minimal fillings of convex bodies.
method Analyzing integral current spaces and proving rigidity properties.
result Convex bodies are the unique minimal fillings of their boundary metrics among integral current spaces and enjoy Lipschitz-volume rigidity.
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and mod…
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
We extend the theory of Patterson-Sullivan measure to any regular covering of a compact manifold using the Busemann compactification and derive an integral formula for the volume entropy. As applications we prove some rigidity theorems for the volume entropy.
The numerical integration plays a fundamental role in understanding the behaviour of many mechanical systems. In this paper some important aspects of the mechanical integrators on the dynamics of a mechanical system are studied. More specific, we have shown that if that the Lie-Trotter integrator is obtained, in case o…
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
The study shows almost maximal volume entropy rigidity for certain manifolds with integral Ricci curvature.
problem Volume entropy rigidity for manifolds with lower integral Ricci curvature bound.
method Analyzing manifolds with specific integral Ricci curvature bounds, diameter, and volume entropy.
result The universal cover of the manifold is close to a hyperbolic space form under certain conditions.
The paper examines rigidity of special submanifolds in spheres with curvature constraints.
problem Rigidity of k-extremal submanifolds in a sphere under curvature conditions. method Proves pinching theorems for submanifolds with various curvature conditions.
result Various curvature conditions lead to rigidity of k-extremal submanifolds. New equations reveal moduli space rigidity in geometric deformations.
problem Understanding moduli space rigidity in geometric deformations.
method Coupled Hitchin-He equations, Lax pair, nonlinear embedding.
result Moduli space is analytically isomorphic to the classical case for small deformations.
The paper proves a quantitative rigidity result for spaces with specific curvature bounds.
problem Understanding the rigidity of spaces with almost maximal volume entropy.
method Analyzing Riemannian manifolds and RCD-spaces with specific curvature conditions. result Spaces with almost maximal volume entropy are closely related to hyperbolic space forms.
By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.
The paper proves rigidity for hypersurfaces with constant shifted curvature functions in warped product manifolds.
problem Characterizing and proving rigidity for hypersurfaces with constant shifted curvature functions.
method Using integral inequalities and Minkowski-type formulas, the paper derives rigidity theorems in sub-static warped product manifolds.
result The paper provides new characterizations and rigidity results for hypersurfaces with constant shifted curvature functions in warped product manifolds.
Revisits Koiso's rigid metrics on complex projective spaces.
problem Computing obstructions to integrability of deformations.
method Elementary complex differential geometry.
result Computes Koiso's obstruction on CPnimesCP1. Paper proves rigidity results for anisotropic capillary hypersurfaces.
problem Rigidity of anisotropic capillary hypersurfaces.
method New Hsiung-Minkowski integral formula for anisotropic capillary hypersurfaces.
result Uniqueness of solution to anisotropic Orlicz-Christoffel-Minkowski problem.
Anosov diffeomorphisms with integrable subbundles have coherent dynamics and spectral rigidity.
problem Characterizing Anosov diffeomorphisms with integrable subbundles.
method Joint integrability of strong stable and unstable subbundles leads to coherent dynamics and spectral rigidity.
result Anosov diffeomorphisms with integrable subbundles are dynamically coherent and have spectral rigidity.
We prove integral rigidity for Seiberg-Witten invariants of 4-manifolds with specific hypersurfaces.
problem Integral rigidity of Seiberg-Witten invariants in 4-manifolds with non-separating hypersurfaces.
method Floer theoretic conditions and interplay between irreducible and reducible solutions to Seiberg-Witten equations.
result Sum of Seiberg-Witten invariants is determined cohomologically for specific 4-manifolds.
Study rigidity on CR Yamabe equation on Sasakian manifolds.
problem Proving rigidity on CR Yamabe equation on Sasakian manifolds.
method Using Jerison-Lee's differential identity and integral estimates.
result Prove that the manifold is CR isometric to Heisenberg group \(\mathbb{H}^n\).
Researchers prove a new inequality for special Riemannian manifolds.
problem Establishing a new integral inequality for a specific class of Riemannian manifolds.
method Developed a Catino-type integral inequality for closed Bach-flat A₂-manifolds.
result Derived rigidity results showing the manifold is either Einstein or a specific product space.
Study on scalar curvature deformations in pseudohermitian manifolds.
problem Deformation of scalar curvature in pseudohermitian manifolds.
method Analogy with Riemannian manifolds, introduction of R-singular spaces, stability conditions, partial infinitesimal rigidity. result Partial infinitesimal rigidity result for scalar curvature of compact pseudohermitian manifolds.
In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying doma…
Elliptic bouquets defined for spin manifolds with circular actions.
problem Defining integration in elliptic cohomology.
method Introducing elliptic bouquets of germs of holomorphic equivariant cohomology classes, integrating them as in K-theory.
result Witten's rigidity theorem follows from integration of elliptic bouquets.
Study classifies 3D Einstein manifolds with cyclic Ricci tensor.
problem Classifying Einstein manifolds with specific tensor properties.
method Derived integral formula involving tensor D for classification.
result Obtained rigidity results for 3D manifolds.
In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, t…
Counterexample shows ADC contact structures can't have isomorphic cohomologies.
problem Rigidity of ADC contact structures and cohomology isomorphisms.
method Provided a counterexample to show non-isomorphic cohomologies.
result ADC contact structures do not have isomorphic integral cohomologies.
Integrates magnetic geodesic and sub-Riemannian flows on Stefel variety, proving integrability and Lax presentations.
problem Integrability of magnetic geodesic and sub-Riemannian flows on Vn,2. method Proves integrability of magnetic geodesic and sub-Riemannian flows on Vn,2 with respect to magnetic field ηdα. result Integrable cases of a heavy rigid body with a gyrostat are derived.
New equations for rigid body motion on infinite-dimensional spaces of operators.
problem Integrating rigid body dynamics on infinite-dimensional spaces of operators.
method Introducing pseudo-Riemannian metrics and adapting classical integrability theory.
result Existence of geodesics and integrals of motion for the rigid body equations.
The paper explores rigidity and splitting theorems for sub-static spaces with minimal hypersurfaces.
problem Rigidity and splitting problems for sub-static systems with boundary.
method Local and global splitting theorems, boundary integral inequalities, Liouville theorem.
result Improvements in rigidity and splitting results for sub-static spaces, including vacuum and non-vacuum cases.
Study on complex Grassmannians' rigidity using Einstein deformations.
problem Characterizing integrable infinitesimal Einstein deformations of complex Grassmannians.
method Analyzing the integrability to second order of infinitesimal deformations using Koiso's obstruction polynomial.
result Characterized integrable deformations as an explicit variety in su(n), showing g is isolated for odd n. Study on biharmonic hypersurfaces in spheres and space forms, proving rigidity under scalar curvature condition.
problem Characterizing biharmonic hypersurfaces in space forms.
method Proved a rigidity result and established an integral formula for biharmonic hypersurfaces.
result Rigidity result under a scalar curvature condition for biharmonic hypersurfaces in space forms.
The study proves a new inequality and formula for manifolds with non-negative Ricci curvature.
problem Proving a sharp mean value inequality for non-negative superharmonic functions.
method Develops a new sharp mean value inequality and an explicit formula for weighted scalar curvature.
result The new inequality removes the radius restriction of Schoen-Yau's result and provides an explicit formula for integral of weighted scalar curvature.
In this paper, we consider some rigidity results for the Einstein metrics as the critical points of some known quadratic curvature functionals on complete manifolds, characterized by some point-wise inequalities. Moreover, we also provide rigidity results by the integral inequalities involving the Weyl curvature, the t…
In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.
The paper explores rigidity and flexibility of isometric extensions with critical Hölder exponent.
problem The critical Hölder exponent in isometric extensions and its implications.
method Convex integration and construction of isometric extensions.
result The Hölder exponent $θ_0= rac12$ is critical, with extensions violating the tangential connection for $θ< rac12$.
Study inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
problem Inequalities involving torsional rigidity and spherical deficit on Riemannian manifolds.
method Establish inequalities and derive an integral identity for a Dirichlet problem.
result Characterize metric balls and measure spherical deficit on Riemannian manifolds.
The paper generalizes Alexandrov theorems for null hypersurfaces with integral curvature conditions.
problem Determining when a submanifold lies on a shear-free null hypersurface under integral curvature conditions.
method Using Minkowski formulas with arbitrary weight to derive rigidity results for submanifolds with weaker integral curvature conditions.
result A necessary and sufficient condition for a submanifold to lie in a shear-free null hypersurface is given by a mean curvature integral inequality.
Characterizes eigenfunctions of Lawson-Osserman cone and proves its integrability.
problem Eigenfunctions and integrability of Lawson-Osserman cone.
method Characterization of eigenfunctions and proof of integrability.
result Lawson-Osserman cone is integrable, generating all Jacobi fields via rotations and translations.
We prove that an n-dimensional, n≥4, compact gradient shrinking Ricci soliton satisfying a L2n-pinching condition is isometric to a quotient of the round Sn, which improves the rigidity theorem given by G. Catino (arXiv:1509.07416vl).
We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding. We study it from the point of view of bihamiltonian geometry. We show how to algebraically integrate it by means of the method of separation of variables.
Study short-time existence of Ricci-DeTurck flow from rough metrics with Morrey-type integrability.
problem Short-time existence of Ricci-DeTurck flow from rough metrics with specific integrability condition.
method Rough existence theory, preservation and improvement of scalar curvature bounds.
result Preservation and improvement of distributional scalar curvature lower bounds under certain conditions.
We find some integral formulas of Simons and Bochner type and use them to study biharmonic and biconservative submanifolds in space forms. We obtain rigidity results that in the biharmonic case represent partial answers to two well-known conjectures on such submanifolds in spheres.
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax the such bound to any finite constant (see Theorem 4.4 for details).