Rigidity Statement Research Articles

Dimension constraints in some problems involving intermediate curvature

In [Comm. Pure Appl. Math. 77 (2024), pp. 441–456] Brendle-Hirsch-Johne proved that T m × S n − m T^m\times S^{n-m} does not admit metrics with positive m m -intermediate curvature when n ≤ 7 n\leq 7 . Chu-Kwong-Lee showed in [Math. Res. Lett., to appear] a corresponding rigidity statement when n ≤ 5 n\leq 5 . In this paper, we show sharpness of the dimension constraints by giving concrete counterexamples in n ≥ 7 n\geq 7 and extending the rigidity result to n = 6 n=6 . Concerning uniformly positive intermediate curvature, we show that simply-connected manifolds with dimension ≤ 5 \leq 5 and bi-Ricci curvature ≥ 1 \geq 1 have finite Urysohn 1-width. Counterexamples are constructed in dimension ≥ 6 \geq 6 .

The Equality Case in the Substatic Heintze–Karcher Inequality

We provide a rigidity statement for the equality case of the Heintze–Karcher inequality in substatic manifolds. We apply such a result in the warped product setting to fully remove assumption (H4) in the celebrated Brendle’s characterization of constant mean curvature hypersurfaces in warped products.

Rigidity of the Torelli subgroup in $\mathrm{Out}(F_{N})$

Let N\ge 4 . We prove that every injective homomorphism from the Torelli subgroup \textup{IA}_{N} to \textup{Out}(F_{N}) differs from the inclusion by a conjugation in \textup{Out}(F_{N}) . This applies more generally to the following subgroups of \textup{Out}(F_{N}) : every finite-index subgroup of \textup{Out}(F_{N}) (recovering a theorem of Farb and Handel); every subgroup of \textup{Out}(F_{N}) that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) ; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of \textup{Out}(F_{N}) –those are subgroups that contain sufficiently many twists in an appropriate sense.Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in \textup{Out}(F_{N}) ; it also implies that all subgroups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) are co-Hopfian.We also prove the same rigidity statement for subgroups of \textup{Out}(F_{3}) which contain a power of every Nielsen transformation. This shows, in particular, that \textup{Out}(F_{3}) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N=3 case.

Positive Mass Theorems for Spin Initial Data Sets With Arbitrary Ends and Dominant Energy Shields

Abstract We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu -|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e., $\textrm{E} < |\textrm{P}|$), there exists a constant $R>0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten’s proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten’s approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

Spectral Torical Band Inequalities and Generalizations of the Schoen–Yau Black Hole Existence Theorem

Abstract Generalized torical band inequalities give precise upper bounds for the width of compact manifolds with boundary in terms of positive pointwise lower bounds for scalar curvature, assuming certain topological conditions. We extend several incarnations of these results in which pointwise scalar curvature bounds are replaced with spectral scalar curvature bounds. More precisely, we prove upper bounds for the width in terms of the principal eigenvalue of the operator $-\Delta +cR$, where $R$ denotes scalar curvature and $c>0$ is a constant. Three separate strategies are employed to obtain distinct results holding in different dimensions and under varying hypotheses, namely we utilize spacetime harmonic functions, $\mu $-bubbles, and spinorial Callias operators. In dimension 3, where the strongest result is produced, we are also able to treat open and incomplete manifolds, and establish the appropriate rigidity statements. Additionally, a version of such spectral torus band inequalities is given where tori are replaced with cubes. Finally, as a corollary, we generalize the classical work of Schoen and Yau, on the existence of black holes due to concentration of matter, to higher dimensions and with alternate measurements of size.

The structure of quasi-complete intersection ideals

We prove that every quasi-complete intersection (q.c.i.) ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by-product we establish a rigidity statement for the minimal two-step Tate complex associated to an ideal I in a local ring R. Furthermore, we define a minimal two-step complete Tate complex T for each ideal I in a local ring R; and prove a rigidity result for it. The complex T is exact if and only if I is a q.c.i. ideal; and in this case, T is the minimal complete resolution of R/I by free R-modules.

Monotonicity Formulas for Harmonic Functions in textrm{RCD}(0,N) Spaces

We generalize to the textrm{RCD}(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in textrm{RCD} spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in textrm{RCD}(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.

Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations

We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3‑dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth ‘spacetime harmonic functions’ in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity statement which asserts that the mass vanishes if and only if the data arise from Minkowski space. The proof has some analogy with both the Witten spinorial approach as well as the marginally outer trapped surface (MOTS) method of Eichmair, Huang, Lee, and Schoen. Furthermore, this paper generalizes the harmonic level set technique used in the Riemannian case by Bray, Stern, and the second and third authors, albeit with a different class of level sets. Thus, even in the time-symmetric (Riemannian) case a new inequality is achieved.

Einstein-Type Structures, Besse’s Conjecture, and a Uniqueness Result for a $$\varphi $$-CPE Metric in Its Conformal Class

In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map \(\varphi : M \rightarrow N\). The resulting system, denoted by (\(\varphi \)-CPE), is natural from the variational viewpoint and describes stationary points for the integrated \(\varphi \)-scalar curvature functional restricted to metrics with unit volume and constant \(\varphi \)-scalar curvature. We prove both a rigidity statement for solutions to (\(\varphi \)-CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (\(\varphi \)-CPE) with \(\varphi \) a harmonic map.

The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets with Toroidal Infinity and Related Rigidity Results

We establish the positive energy theorem and a Penrose-type inequality for 3-dimensional asymptotically hyperboloidal initial data sets with toroidal infinity, weakly trapped boundary, and satisfying the dominant energy condition. In the umbilic case, a rigidity statement is proven showing that the total energy vanishes precisely when the initial data manifold is isometric to a portion of the canonical slice of the associated Kottler spacetime. Furthermore, we provide a new proof of the recent rigidity theorems of Eichmair-Galloway-Mendes [10] in dimension 3, with weakened hypotheses in certain cases. These results are obtained through an analysis of the level sets of spacetime harmonic functions.

Deformations of Lagrangian submanifolds in log-symplectic manifolds

This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that the gauge equivalence relation of the DGLA corresponds with the geometric notion of equivalence by Hamiltonian isotopies. We discuss the corresponding moduli space, and we prove a rigidity statement for the more flexible equivalence relation by Poisson isotopies.

A Talenti-type comparison theorem for {{,mathrm{RCD},}}(K,N) spaces and applications

We prove pointwise and L^{p}-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more precisely of an {{,mathrm{RCD},}}(K,N) metric measure space, with K>0 and Nin (1,infty )). The obtained Talenti-type comparison is sharp, rigid and stable with respect to L^{2}/measured-Gromov–Hausdorff topology; moreover, several aspects seem new even for smooth Riemannian manifolds. As applications of such Talenti-type comparison, we prove a series of improved Sobolev-type inequalities, and an {{,mathrm{RCD},}} version of the St. Venant-Pólya torsional rigidity comparison theorem (with associated rigidity and stability statements). Finally, we give a probabilistic interpretation (in the setting of smooth Riemannian manifolds) of the aforementioned comparison results, in terms of exit time from an open subset for the Brownian motion.

The Arakelov-Zhang pairing and Julia sets

The Arakelov-Zhang pairing ⟨ ψ , ϕ ⟩ \langle \psi ,\phi \rangle is a measure of the “dynamical distance” between two rational maps ψ \psi and ϕ \phi defined over a number field K K . It is defined in terms of local integrals on Berkovich space at each completion of K K . We obtain a simple expression for the important case of the pairing with a power map, written in terms of integrals over Julia sets. Under certain disjointness conditions on Julia sets, our expression simplifies to a single canonical height term; in general, this term is a lower bound. As applications of our method, we give bounds on the difference between the canonical height h ϕ h_\phi and the standard Weil height h h , and we prove a rigidity statement about polynomials that satisfy a strong form of good reduction.

Stability of a quasi-local positive mass theorem for graphical hypersurfaces of Euclidean space

We present a quasi-local version of the stability of the positive mass theorem. We work with the Brown–York quasi-local mass as it possesses positivity and rigidity properties, and therefore the stability of this rigidity statement can be studied. Specifically, we ask if the Brown–York mass of the boundary of some compact manifold is close to zero, must the manifold be close to a Euclidean domain in some sense? Here we consider a class of compact n n -manifolds with boundary that can be realized as graphs in R n + 1 \mathbb {R}^{n+1} , and establish the following. If the Brown–York mass of the boundary of such a compact manifold is small, then the manifold is close to a Euclidean hyperplane with respect to the Federer–Fleming flat distance.

Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

AbstractIn this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper, we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in \(L^p\), \(1\le p<2\). In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.

The Penrose Inequality and Positive Mass Theorem with Charge for Manifolds with Asymptotically Cylindrical Ends

We establish the charged Penrose inequality for time symmetric initial data sets having an outermost minimal surface boundary and finitely many asymptotically cylindrical ends, with an appropriate rigidity statement. This is accomplished by a doubling argument based on the work of Weinstein and Yamada, and a subsequent application of the ordinary charged Penrose inequality as established by Khuri, Weinstein, and Yamada. Furthermore, the techniques used in the aforementioned proof allow for the proof of the positive mass theorem with charge for such manifolds.

On the Mass of Static Metrics with Positive Cosmological Constant: II

This is the second of two works, in which we discuss the definition of an appropriate notion of mass for static metrics, in the case where the cosmological constant is positive and the model solutions are compact. In the first part, we have established a positive mass statement, characterising the de Sitter solution as the only static vacuum metric with zero mass. In this second part, we prove optimal area bounds for horizons of black hole type and of cosmological type, corresponding to Riemannian Penrose inequalities and to cosmological area bounds à la Boucher–Gibbons–Horowitz, respectively. Building on the related rigidity statements, we also deduce a uniqueness result for the Schwarzschild–de Sitter spacetime.

A Penrose-type inequality with angular momentum and charge for axisymmetric initial data

A lower bound for the ADM mass is established in terms of angular momentum, charge, and horizon area in the context of maximal, axisymmetric initial data for the Einstein-Maxwell equations which satisfy the weak energy condition. If, on the horizon, the given data agree to a certain extent with the associated model Kerr-Newman data, then the inequality reduces to the conjectured Penrose inequality with angular momentum and charge. In addition, a rigidity statement is also proven whereby equality is achieved if and only if the data set arises from the canonical slice of a Kerr-Newman spacetime.

ON ABSTRACT HOMOMORPHISMS OF CHEVALLEY GROUPS OVER THE COORDINATE RINGS OF AFFINE CURVES

The goal of this paper is to establish a general rigidity statement for abstract representations of elementary subgroups of Chevalley groups of rank ≥ 2 over a class of commutative rings that includes the localizations of 1-generated rings and the coordinate rings of affine curves. Our main result implies, for example, that any finite-dimensional representation of SLn(ℤ[X]) (n ≥ 3) over an algebraically closed field of characteristic zero has a standard description, yielding thereby the first unconditional rigidity statement for finitely generated linear groups other than arithmetic groups/lattices.