An official website of the United States government
Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres.
more »
« less
Endperiodic automorphisms of surfaces and foliations
We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations $${\mathcal{F}}$$ and $${\mathcal{F}}^{\prime }$$ are transverse to a common one-dimensional foliation $${\mathcal{L}}$$ whose monodromy on the non-compact leaves of $${\mathcal{F}}$$ exhibits the nice dynamics of Handel–Miller theory, then $${\mathcal{L}}$$ also induces monodromy on the non-compact leaves of $${\mathcal{F}}^{\prime }$$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.
more »
« less
- Award ID(s):
- 2054909
- PAR ID:
- 10324067
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- Volume:
- 41
- Issue:
- 1
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 66 to 212
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.more » « less
-
The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.more » « less
-
We give an axiomatic characterization of multiple Dirichlet series over the function field Fq(T), generalizing a set of axioms given by Diaconu and Pasol. The key axiom, relating the coefficients at prime powers to sums of the coefficients, formalizes an observation of Chinta. The existence of multiple Dirichlet series satisfying these axioms is proved by exhibiting the coefficients as trace functions of explicit perverse sheaves and using properties of perverse sheaves. The multiple Dirichlet series defined this way include, as special cases, many that have appeared previously in the literature.more » « less
-
A bstract We study electric-magnetic duality in compactifications of M-theory on twisted connected sum (TCS) G 2 manifolds via duality with F-theory. Specifically, we study the physics of the D3-branes in F-theory compactified on a Calabi-Yau fourfold Y , dual to a compactification of M-theory on a TCS G 2 manifold X . $$ \mathcal{N} $$ N = 2 supersymmetry is restored in an appropriate geometric limit. In that limit, we demonstrate that the dual of D3-branes probing seven-branes corresponds to the shrinking of certain surfaces and curves, yielding light particles that may carry both electric and magnetic charges. We provide evidence that the Minahan-Nemeschansky theories with E n flavor symmetry may be realized in this way. The SL(2 , ℤ) monodromy of the 3/7-brane system is dual to a Fourier-Mukai transform of the dual IIA/M-theory geometry in this limit, and we extrapolate this monodromy action to the global compactification. Away from the limit, the theory is broken to $$ \mathcal{N} $$ N = 1 supersymmetry by a D-term.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2019.56
