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The Kepler problem is the special case $\alpha = 1$ of the power law problem:to solve Newton's equations for a central force whose potential is of the form$-\mu/r^{\alpha}$ where $\mu$ is a coupling constant. Associated to such aproblem is a two-dimensional cone with cone angle $2 \pi c$ with $c = 1 -\frac{\alpha}{2}$. We construct a transformation taking the geodesics of thiscone to the zero energy solutions of the $\alpha$-power law problem. The`Kepler Cone' is the cone associated to the Kepler problem. This zero-energycone transformation is a special case of a transformation discovered byMaclaurin in the 1740s transforming the $\alpha$- power law problem for anyenergies to a `Maclaurin dual' $\gamma$-power law problem where $\gamma =\frac{2 \alpha}{2-\alpha}$ and which, in the process, mixes up the energy ofone problem with the coupling constant of the other. We derive Maclaurinduality using the Jacobi-Maupertuis metric reformulation of mechanics. We thenuse the conical metric to explain properties of Rutherford-type scattering offpower law potentials at positive energies. The one possibly new result in thepaper concerns ``star-burst curves'' which arise as limits of families negativeenergy solutions as their angular momentum tends to zero. We relate geodesicscattering on the cone to Rutherford type scattering of beams of solutions inthe potential. We describe some history around Maclaurin duality and give twoderivations of the Jacobi-Maupertuis metric reformulation of classicalmechanics. The piece is expository, aimed at an upper-division undergraduate.Think American Math. Monthly.

We prove nonemptyness of domains of proper discontinuity of Anosov groups ofaffine Lorentzian transformations.

As in the case of irreducible holomorphic symplectic manifolds, the perioddomain $Compl$ of compact complex tori of even dimension $2n$ contains twistorlines. These are special $2$-spheres parametrizing complex tori whose complexstructures arise from a given quaternionic structure. In analogy with the caseof irreducible holomorphic symplectic manifolds, we show that the periods ofany two complex tori can be joined by a {\em generic} chain of twistor lines.We also prove a criterion of twistor path connectivity of loci in $Compl$ wherea fixed second cohomology class stays of Hodge type (1,1). Furthermore, we showthat twistor lines are holomorphic submanifolds of $Compl$, of degree $2n$ inthe Pl\"ucker embedding of $Compl$.

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result [Hezari et al., Ann. Henri Poincare 18(12), 3759-3792 (2017)], which was only concerned with the Neumann Laplace spectrum.

This paper is the third and final component of a three-part effort onobservers contracting a Riemannian distance between the state of the system andits estimate. In Part I, we showed that such a contraction property holds ifthe system dynamics and the Riemannian metric satisfy two key conditions: adifferential detectability property and a geodesic monotonicity property. Withthe former condition being the focus of Part II, in this Part III, we study thelatter condition in relationship to the nullity of the second fundamental formof the output function. We formulate sufficient and necessary conditions for itto hold. We establish a link between it and the infinite gain margin property,and we provide a systematic way for constructing a metric satisfying thiscondition. Finally, we illustrate cases where both conditions hold and proposeways to facilitate the satisfaction of these two conditions together.

Let Γ be a discrete group of isometries acting on the complex hyperbolic n-space HCn. In this note, we prove that if Γ is convex-cocompact, torsion-free, and the critical exponent δ(Γ) is strictly lesser than 2, then the complex manifold HCn/Γ is Stein. We also discuss several related conjectures.

We relate the sub-Riemannian geometry on the group of rigid motions of theplane to `bicycling mathematics'. We show that this geometry's geodesicscorrespond to bike paths whose front tracks are either non-inflectional Eulerelasticae or straight lines, and that its infinite minimizing geodesics (or`metric lines') correspond to bike paths whose front tracks are either straightlines or `Euler's solitons' (also known as Syntractrix or Convicts' curves).

Geometrically infinite Kleinian groups have non-conical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the non-conical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the non-conical limit set is positive.

We show that it is $\mathsf{NP}$-hard to approximate the hypersphericalradius of a triangulated manifold up to an almost-polynomial factor.

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has arationalization $Y \to Y_{(0)}$ which is well-understood. Early on it was foundthat the induced map $[X,Y] \to [X,Y_{(0)}]$ on sets of mapping classes isfinite-to-one. The sizes of the preimages need not be bounded; we show,however, that as the complexity (in a suitable sense) of a rational mappingclass increases, these sizes are at most polynomial. This ``torsion''information about $[X,Y]$ is in some sense orthogonal to rational homotopytheory but is nevertheless an invariant of the rational homotopy type of $Y$ inat least some cases. The notion of complexity is geometric and we also prove aconjecture of Gromov \cite{GrMS} regarding the number of mapping classes thathave Lipschitz constant at most $L$.

We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size 1\14 These improve the earlier results in the subject for smooth potentials, where an expansion exists in a 7kappa; neighborhood of the diagonal. We obtain our results by finding upper bounds of the form Cmm2 for the Bergman coefficients bm(x,) which is an interesting problem on its own. We find such upper bounds using the method of [3]. We also show that sharpening these upper bounds would improve the rate of shrinking neighborhoods of the diagonal x = y in our results. In the special case of metrics with local constant holomorphic sectional curvatures, we obtain off-diagonal asymptotic in a fixed (ask→∞) neighborhood of the diagonal, which recovers a result of Berman [2] (see Remark 3.5 of [2] for higher dimensions). In this case, we also find an explicit formula for the Bergman kernel mod $O(e-kδ) $.

In this paper, we establish a min-max theory for constructing minimal diskswith free boundary in any closed Riemannian manifold. The main result is aneffective version of the partial Morse theory for minimal disks with freeboundary established by Fraser. Our theory also includes as a special case themin-max theory for Plateau problem of minimal disks, which can be used togeneralize the famous work by Morse-Thompkins and Shiffman on minimal surfacesin $\mathbf{R}^n$ to the Riemannian setting. More precisely, we generalize the min-max construction of minimal surfacesusing harmonic replacement introduced by Colding and Minicozzi to the freeboundary setting. As a key ingredient to this construction, we show an energyconvexity for weakly harmonic maps with mixed Dirichlet and free boundariesfrom the half unit $2$-disk in $\mathbf{R}^2$ into any closed Riemannianmanifold, which in particular yields the uniqueness of such weakly harmonicmaps. This is a free boundary analogue of the energy convexity and uniquenessfor weakly harmonic maps with Dirichlet boundary on the unit $2$-disk proved byColding and Minicozzi.

In the 1970s and again in the 1990s, Gromov gave a number of theorems andconjectures motivated by the notion that the real homotopy theory of compactmanifolds and simplicial complexes influences the geometry of maps betweenthem. The main technical result of this paper supports this intuition: we showthat maps of differential algebras are closely shadowed, in a technical sense,by maps between the corresponding spaces. As a concrete application, we provethe following conjecture of Gromov: if $X$ and $Y$ are finite complexes with$Y$ simply connected, then there are constants $C(X,Y)$ and $p(X,Y)$ such thatany two homotopic $L$-Lipschitz maps have a $C(L+1)^p$-Lipschitz homotopy (andif one of the maps is a constant, $p$ can be taken to be $2$.) We hope that itwill lead more generally to a better understanding of the space of maps from$X$ to $Y$ in this setting.

Consider a compact Riemannian manifold of dimension ≥ 3 with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension ≥ 3 having nonnegative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.

These are lectures on discrete groups of isometries of complex hyperbolicspaces, aimed to discuss interactions between the function theory on complexhyperbolic manifolds and the theory of discrete groups.

We show that the space of metrics of positive scalar curvature on any3-manifold is either empty or contractible. Second, we show that thediffeomorphism group of every 3-dimensional spherical space form deformationretracts to its isometry group. This proves the Generalized Smale Conjecture.Our argument is independent of Hatcher's theorem in the $S^3$ case and inparticular it gives a new proof of the $S^3$ case.

With the help of a generalization of the Fermat principle in generalrelativity, we show that chains in CR geometry are geodesics of a certainKropina metric constructed from the CR structure. We study the projectiveequivalence of Kropina metrics and show that if the kernel distributions of thecorresponding 1-forms are non-integrable then two projectively equivalentmetrics are trivially projectively equivalent. As an application, we show thatsufficiently many chains determine the CR structure up to conjugacy,generalizing and reproving the main result of [J.-H. Cheng, 1988]. Thecorrespondence between geodesics of the Kropina metric and chains allows us touse the methods of metric geometry and the calculus of variations to studychains. We use these methods to re-prove the result of [H. Jacobowitz, 1985]that locally any two points of a strictly pseudoconvex CR manifolds can bejoined by a chain. Finally, we generalize this result to the global setting byshowing that any two points of a connected compact strictly pseudoconvex CRmanifold which admits a pseudo-Einstein contact form with positiveTanaka-Webster scalar curvature can be joined by a chain.

We extend several notions and results from the classical Patterson-Sullivantheory to the setting of Anosov subgroups of higher rank semisimple Lie groups,working primarily with invariant Finsler metrics on associated symmetricspaces. In particular, we prove the equality between the Hausdorff dimensionsof flag limit sets, computed with respect to a suitable Gromov (pre-)metric onthe flag manifold, and the Finsler critical exponents of Anosov subgroups.

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among3-dimensional $\kappa$-solutions. In this paper, we present an alternativeproof for this fact and show that compact $\kappa$-solutions are rotationalsymmetric. Our proof arose from independent work relating to our StrongStability Theorem for singular Ricci flows.

We prove a Morse lemma for regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings. We apply it to give a new coarse geometric characterization of Anosov subgroups of the isometry groups of such spaces simply as undistorted subgroups which are uniformly regular.

In this paper, we establish compactness results of some class of conformallycompact Einstein 4-manifolds. In the first part of the paper, we improve theearlier results obtained by Chang-Ge. In the second part of the paper, asapplications, we derive some compactness results under perturbation conditionswhen the L^2-norm of the Weyl curvature is small. We also derive the globaluniqueness of conformally compact Einstein metrics on the 4-Ball constructed inthe earlier work of Graham-Lee.

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature (n- 1 , 1) and trans-Lorentzian signature (n- 2 , 2).

For a given null-cobordant Riemannian n n -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n n . In the appendix the bound is improved to one that is O ( L 1 + ε ) O(L^{1+\varepsilon }) for every ε > 0 \varepsilon >0 .This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y Y is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic L L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y .

Answering a question by Margulis we prove that the conclusion of Selberg'sLemma fails for discrete isometry groups of negatively curved Hadamardmanifolds.

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

We propose several common extensions of the classes of Anosov subgroups andgeometrically finite Kleinian groups among discrete subgroups of semisimple Liegroups. We relativize various dynamical and coarse geometric characterizationsof Anosov subgroups given in our earlier work, extending the class fromintrinsically hyperbolic to relatively hyperbolic subgroups. We proveimplications and equivalences between the various relativizations.

© 2018, Mathematical Sciences Publishers. All Rights reserved. We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces X/Γ for arbitrary discrete subgroups Γ

We study Riemannian coverings [Formula presented] where M~ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations [Formula presented] are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings [Formula presented] are the universal Riemannian coverings of spherical space forms. When M=G/K is reasonably well understood, in particular when G/K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M~. In other words we show that [Formula presented] is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M~ we work out the full isometry group of M~ extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

We study Riemannian coverings [Formula presented] where M~ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations [Formula presented] are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings [Formula presented] are the universal Riemannian coverings of spherical space forms. When M=G/K is reasonably well understood, in particular when G/K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M~. In other words we show that [Formula presented] is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M~ we work out the full isometry group of M~ extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

We prove that the boundary of the future of a surface K consists precisely of the points p that lie on a null geodesic orthogonal to K such that between K and p there are no points conjugate to K nor intersections with another such geodesic. Our theorem has applications to holographic screens and their associated light sheets and in particular enters the proof that holographic screens satisfy an area law.

In this article we study the short-time existence of conformal Ricci flow onasymptotically hyperbolic manifolds. We also prove a local Shi's type curvaturederivative estimate for conformal Ricci flow.

In this paper, we are concerned with the problem of creating flattening maps of simply connected open surfaces in ℝ3Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing maps with any prescribed density distribution. By varying the initial density distribution, a large variety of flattening maps with different properties can be achieved. For instance, area-preserving parameterizations of simply connected open surfaces can be easily computed. Experimental results are presented to demonstrate the effectiveness of our proposed method. Applications to data visualization and surface remeshing are explored.

We complete the proof of the Generalized Smale Conjecture, apart from thecase of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic3-manifolds. We use an approach based on Ricci flow through singularities,which applies uniformly to spherical space forms other than $S^3$ and $RP^3$and hyperbolic manifolds, to prove that the moduli space of metrics of constantsectional curvature is contractible. As a corollary, for such a 3-manifold $X$,the inclusion $\text{Isom} (X,g)\to \text{Diff}(X)$ is a homotopy equivalencefor any Riemannian metric $g$ of constant sectional curvature.

© 2017, Springer International Publishing AG. We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture “rank one behavior” of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with R2. As an application, we show that positively curved metrics on S3 and RP3 with almost maximal width must be nearly round.

We verify a conjecture of Perelman, which states that there exists acanonical Ricci flow through singularities starting from an arbitrary compactRiemannian 3-manifold. Our main result is a uniqueness theorem for such flows,which, together with an earlier existence theorem of Lott and the second namedauthor, implies Perelman's conjecture. We also show that this flow throughsingularities depends continuously on its initial condition and that it may beobtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of threemanifolds --- in particular to the Generalized Smale Conjecture --- which willappear in a subsequent paper.

In [2], the authors develop a global correspondence between immersed weaklyhorospherically convex hypersurfaces $\phi:M^n \to \mathbb{H}^{n+1}$ and aclass of conformal metrics on domains of the round sphere $\mathbb{S}^n$. Someof the key aspects of the correspondence and its consequences have dimensionalrestrictions $n\geq3$ due to the reliance on an analytic proposition from [5]concerning the asymptotic behavior of conformal factors of conformal metrics ondomains of $\mathbb{S}^n$. In this paper, we prove a new lemma about theasymptotic behavior of a functional combining the gradient of the conformalfactor and itself, which allows us to extend the global correspondence andembeddedness theorems of [2] to all dimensions $n\geq2$ in a unified way. Inthe case of a single point boundary $\partial_{\infty}\phi(M)=\{x\} \subset\mathbb{S}^n$, we improve these results in one direction. As an immediateconsequence of this improvement and the work on elliptic problems in [2], wehave a new, stronger Bernstein type theorem. Moreover, we are able to extendthe Liouville and Delaunay type theorems from [2] to the case of surfaces in$\mathbb{H}^{3}$.

Based on properties of n-subharmonic functions we show that a complete,noncompact, properly embedded hypersurface with nonnegative Ricci curvature inhyperbolic space has an asymptotic boundary at infinity of at most two points.Moreover, the presence of two points in the asymptotic boundary is a rigiditycondition that forces the hypersurface to be an equidistant hypersurface abouta geodesic line in hyperbolic space. This gives an affirmative answer to thequestion raised by Alexander and Currier in 1990.

In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blow-up method to derive curvature estimates for conformally compact Einstein manifolds with large renormalized volume. The major part of this paper is the study of how a property of the conformal infinity influences the geometry of the interior of a conformally compact Einstein manifold. Specifically we are interested in conformally compact Einstein manifolds with conformal infinity whose Yamabe invariant is close to that of the round sphere. Based on the approach initiated by Dutta and Javaheri we present a complete proof of the relative volume inequality \[ ( Y ( ∂ X , [ g ^ ] ) Y ( S n − 1 , [ g S ] ) ) n − 1 2 ≤ V o l ( ∂ B g + ( p , t ) ) V o l ( ∂ B g H ( 0 , t ) ) ≤ V o l ( B g + ( p , t ) ) V o l ( B g H ( 0 , t ) ) ≤ 1 , \left (\frac {Y(\partial X, [\hat {g}])}{Y(\mathbb {S}^{n-1}, [g_{\mathbb {S}}])}\right )^{\frac {n-1}{2}}\leq \frac {Vol(\partial B_{g^+}(p, t))} {Vol(\partial B_{g_{\mathbb {H}}}(0, t))} \leq \frac {Vol(B_{g^+}(p, t))} {Vol(B_{g_{\mathbb {H}}}(0, t))}\leq 1, \] for conformally compact Einstein manifolds. This leads not only to the complete proof of the rigidity theorem for conformally compact Einstein manifolds in arbitrary dimension without spin assumption but also a new curvature pinching estimate for conformally compact Einstein manifolds with conformal infinities having large Yamabe invariant. We also derive curvature estimates for such manifolds.

This survey is based on a series of lectures that we gave at MSRI in Spring2015 and on a series of papers, mostly written jointly with Joan Porti. Ourgoal here is to: 1. Describe a class of discrete subgroups $\Gamma

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs (M,ξ) where M = G/H is a Riemannian normal homogeneous space, G is a compact simple Lie group, and ξ ϵ g defines a nonzero Killing vector field of constant length on M. The method there was direct computation. Here we make use of the moment map M → g and the flag manifold structure of Ad (G)ξ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where ξ is elliptic and G is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.

In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author’s 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if Γ \ M is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold M, then Γ \ M is homogeneous if and only if each isometry γ∈ Γ is of constant displacement. The description of bounded isometries in this paper gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford’s use of quaternions in non-Euclidean geometry, and we sketch that in a historical note.

For a surface in 3-sphere, by identifying the conformal round 3-sphere as theprojectivized positive light cone in Minkowski 5-spacetime, we use theconformal Gauss map and the conformal transform to construct the associatehomogeneous 4-surface in Minkowski 5-spacetime. We then derive the localfundamental theorem for a surface in conformal round 3-sphere from that of theassociate 4-surface in Minkowski 5-spacetime. More importantly, following theidea of Fefferman and Graham, we construct local scalar invariants for asurface in conformal round 3-sphere. One distinct feature of our constructionis to link the classic work of Blaschke to the works of Bryan andFefferman-Graham.

Let G be a complex simple direct limit group, specifically SL(∞; C) , SO(∞; C) or Sp(∞; C). Let F be a (generalized) flag in C∞. If G is SO(∞; C) or Sp(∞; C) we suppose further that F is isotropic. Let Z denote the corresponding flag manifold; thus Z= G/ Q where Q is a parabolic subgroup of G. In a recent paper (Penkov and Wolf in Real group orbits on flag ind-varieties of SL∞(C) , to appear in Proceedings in Mathematics and Statistics) we studied real forms G0 of G and properties of their orbits on Z. Here we concentrate on open G0-orbits D⊂ Z. When G0 is of hermitian type we work out the complete G0-orbit structure of flag manifolds dual to the bounded symmetric domain for G0. Then we develop the structure of the corresponding cycle spaces MD. Finally we study the real and quaternionic analogs of these theories. All this extends results from the finite-dimensional cases on the structure of hermitian symmetric spaces and cycle spaces (in chronological order: Wolf in Bull Am Math Soc 75:1121–1237, 1969; Wolf et al. in Ann Math 105:397–448, 1977; Wolf in Ann Math 136:541–555, 1992; Wolf in Compact subvarieties in flag domains, 1994; Wolf and Zierau in Math Ann 316:529–545, 2000; Huckleberry et al. in Journal für die reine und angewandte Mathematik 2001:171–208, 2001; Huckleberry and Wolf in Cycle spaces of real forms of SLn(C) , Springer, New York, pp 111–133, 2002; Wolf and Zierau in J Lie Theory 13:189–191, 2003; Huckleberry and Wolf in Ann Scuola Norm Sup Pisa Cl Sci (5) 9:573-580, 2010).

© 2016, Springer Science+Business Media New York. In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups.

We develop a general regulated volume expansion for the volume of a manifold with boundary whose measure is suitably singular along a separating hypersurface. The expansion is shown to have a regulator independent anomaly term and a renormalized volume term given by the primitive of an associated anomaly operator. These results apply to a wide range of structures. We detail applications in the setting of measures derived from a conformally singular metric. In particular, we show that the anomaly generates invariant (Q-curvature, transgression)-type pairs for hypersurfaces with boundary. For the special case of anomalies coming from the volume enclosed by a minimal hypersurface ending on the boundary of a Poincare--Einstein structure, this result recovers Branson's Q-curvature and corresponding transgression. When the singular metric solves a boundary version of the constant scalar curvature Yamabe problem, the anomaly gives generalized Willmore energy functionals for hypersurfaces with boundary. Our approach yields computational algorithms for all the above quantities, and we give explicit results for surfaces embedded in 3-manifolds.

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem. Using existence results for asymptotic solutions to this problem, we develop the details of how to proliferate conformal hypersurface invariants. In addition we show how to compute the the solution's asymptotics. We also develop a calculus of conformal hypersurface invariant differential operators and in particular, describe how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we develop new higher dimensional analogues of the Willmore energy for embedded surfaces. This complements recent progress on the existence and construction of such functionals.

Killing vector fields of constant length correspond to isometries of constant displacement. Those in turn have been used to study homogeneity of Riemannian and Finsler quotient manifolds. Almost all of that work has been done for group manifolds or, more generally, for symmetric spaces. This paper extends the scope of research on constant length Killing vector fields to a class of Riemannian normal homogeneous spaces.