Rigidity Theorem Research Articles

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

Rigidity Theorem for Self-Affine Arcs

Rigidity Theorem for Self-Affine Arcs

Scheiderer motives and equivariant higher topos theory

We give an algebro-geometric interpretation of C2-equivariant stable homotopy theory by means of the b-topology introduced by Claus Scheiderer in his study of 2-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a ∞-topos X with G-action a presentable stable ∞-category SpG(X), which recovers the ∞-category SpG of genuine G-spectra when X is the terminal G-∞-topos. Given a scheme X with 12∈OX, our construction then specializes to produce an ∞-category SpbC2(X) of “b-sheaves with transfers” as b-sheaves of spectra on the small étale site of X equipped with certain transfers along the extension X[i]→X; if X is the spectrum of a real closed field, then SpbC2(X) recovers SpC2. On a large class of schemes, we prove that, after p-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the b-variant SHb(X) of the ∞-category SH(X) of motivic spectra over X (in the sense of Morel-Voevodsky), and produce a natural equivalence of ∞-categories SHb(X)p∧≃SpbC2(X)p∧ through amalgamating the étale and real étale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the C2-Tate construction, which may be of independent interest. Finally, as applications, we deduce a “b-rigidity” theorem, use the Segal conjecture to show étale descent of the 2-complete b-motivic sphere spectrum, and construct a parametrized version of the C2-Betti realization functor of Heller-Ormsby.

Conformal infinitesimal variations of submanifolds

This paper belongs to the realm of conformal geometry and deals with Euclidean submanifolds that admit smooth variations that are infinitesimally conformal. Conformal variations of Euclidean submanifolds are a classical subject in differential geometry. In fact, already in 1917 Cartan classified parametrically the Euclidean hypersurfaces that admit nontrivial conformal variations. Our first main result is a Fundamental theorem for conformal infinitesimal variations. The second is a rigidity theorem for Euclidean submanifolds that lie in low codimension.

𝐾-theory and topological cyclic homology of henselian pairs

Given a henselian pair ( R , I ) (R, I) of commutative rings, we show that the relative K K -theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace K → T C K \to \mathrm {TC} . This yields a generalization of the classical Gabber–Gillet–Thomason–Suslin rigidity theorem (for mod n n coefficients, with n n invertible in R R ) and McCarthy’s theorem on relative K K -theory (when I I is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between p p -adic K K -theory and topological cyclic homology for a large class of p p -adic rings. In addition, we show that K K -theory with finite coefficients satisfies continuity for complete noetherian rings which are F F -finite modulo p p . Our main new ingredient is a basic finiteness property of T C \mathrm {TC} with finite coefficients.

Integral inequalities for holomorphic maps and applications

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the Kahler-Ricci flow. The equality case is characterized for some special settings.

A Rigidity Theorem for the deformed Hermitian-Yang-Mills equation

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant C such that $$\begin{aligned} -\frac{1}{C}\omega<\sqrt{-1}F<C\omega . \end{aligned}$$ We also consider the self-shrinker over $$\mathbb {C}^n$$ to the corresponding parabolic flow. We prove that the self-shrinker over $${\mathbb {C}}^n$$ is a quadratic polynomial function. At last, we show a similar rigidity theorem for the J-equations and the self-shrinkers over $${\mathbb {C}}^n$$ to J-flow.

A rigidity theorem for Killing vector fields on compact manifolds with almost nonpositive Ricci curvature

We prove a rigidity theorem for Killing vector fields of a manifold with almost nonpositive Ricci curvature, which is a generalization of Bochner’s classical results.

Codimension bounds and rigidity of ancient mean curvature flows by the tangent flow at −∞

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, by adapting the work of Colding–Minicozzi[11], we prove codimension bounds for ancient mean curvature flows by their tangent flow at [Formula: see text]. In the case of the [Formula: see text]-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at [Formula: see text].

The Fundamental and Rigidity Theorems for Pseudohermitian Submanifolds in the Heisenberg Groups

In this paper, we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups. In particular, we obtain the uniqueness and existence theorems, and some rigidity theorems.

Some rigidity theorems on Finsler manifolds

&lt;abstract&gt;&lt;p&gt;We prove that, for a Finsler manifold with the weighted Ricci curvature bounded below by a positive number, it is a Finsler sphere if and only if the diam attains its maximal value, if and only if the volume attains its maximal value, and if and only if the first closed eigenvalue of the Finsler-Laplacian attains its lower bound. These generalize some rigidity theorems in Riemannian geometry to the Finsler setting.&lt;/p&gt;&lt;/abstract&gt;

Rigidity theorems for Hénon maps—II

Rigidity theorems for Hénon maps—II

Local rigidity, contact homeomorphisms, and conformal factors

We show that if the image of a Legendrian submanifold under a contact homeomorphism (i.e. a homeomorphism that is a $C^0$-limit of contactomorphisms) is smooth then it is Legendrian, assuming only positive local lower bounds on the conformal factors of the approximating contactomorphisms. More generally the analogous result holds for coisotropic submanifolds in the sense of arXiv:1306.6367. This is a contact version of the Humili\`ere-Leclercq-Seyfaddini coisotropic rigidity theorem in $C^0$ symplectic geometry, and the proof adapts the author's recent re-proof of that result in arXiv:1912.13043 based on a notion of local rigidity of points on locally closed subsets. We also provide two different flavors of examples showing that a contact homeomorphism can map a submanifold that is transverse to the contact structure to one that is smooth and tangent to the contact structure at a point.

The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the N\'eron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow's $K/k$-trace functor. As an application of the Hodge Index Theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and Yuan-Zhang.

Discontinuity of Straightening in Anti-Holomorphic Dynamics: II

Abstract In [21], Milnor found Tricorn-like sets in the parameter space of real cubic polynomials. We give a rigorous definition of these Tricorn-like sets as suitable renormalization loci and show that the dynamically natural straightening map from such a Tricorn-like set to the original Tricorn is discontinuous. We also prove some rigidity theorems for polynomial parabolic germs, which state that one can recover unicritical holomorphic and anti-holomorphic polynomials from their parabolic germs.

Variational Problems of Surfaces in a Sphere

Let $$x:M \to \mathbb{S}{^{n + p}}(1)$$ be an n-dimensional submanifold immersed in an (n + p)-dimensional unit sphere $$\mathbb{S}{^{n + p}}(1)$$ . In this paper, we study n-dimensional submanifolds immersed in $$\mathbb{S}{^{n + p}}(1)$$ which are critical points of the functional $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ , where S is the squared length of the second fundamental form of the immersion x. When $$x:M \to \mathbb{S}{^{2 + p}}(1)$$ is a surface in $$\mathbb{S}{^{2 + p}}(1)$$ , the functional $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ represents double volume of image of Gaussian map. For the critical surface of $${\cal S}(x)$$ , we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic. Furthermore, we establish a rigidity theorem for the critical surface of $${\cal S}(x)$$ .

Condensed Ricci curvature of complete and strongly regular graphs

We study a modified notion of Ollivier's coarse Ricci curvature on graphs introduced by Lin, Lu, and Yau in [11]. We establish a rigidity theorem for complete graphs that shows a connected finite simple graph is complete if and only if the Ricci curvature is strictly greater than one. We then derive explicit Ricci curvature formulas for strongly regular graphs in terms of the graph parameters and the size of a maximal matching in the core neighborhood. As a consequence we are able to derive exact Ricci curvature formulas for strongly regular graphs of girth 4 and 5 using elementary methods. An example is provided that shows there is no exact formula for the Ricci curvature for strongly regular graphs of girth $3$ that is purely in terms of graph parameters.

Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

AbstractLet $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity.First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type.Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

Uncertainty principle and its rigidity on complete gradient shrinking Ricci solitons

We prove rigidity theorems for shrinking gradient Ricci solitons supporting the Heisenberg-Pauli-Weyl uncertainty principle with the sharp constant in R n \mathbb {R}^n . In addition, we partially give analogous rigidity results of the Caffarelli-Kohn-Nirenberg inequalities on shrinking Ricci solitons.

Symmetries of 3-polytopes with fixed edge lengths

We consider an interesting class of combinatorial symmetries of polytopes which we call \emph{edge-length preserving combinatorial symmetries}. These symmetries not only preserve the combinatorial structure of a polytope but also map each edge of the polytope to an edge of the same length. We prove a simple sufficient condition for a polytope to realize all edge-length preserving combinatorial symmetries by isometries of ambient space. The proof of this condition uses Cauchy's rigidity theorem in an unusual way.