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We show that the near horizon geometry of 5-dimensional extreme (i.e., degenerate) stationary vacuum black holes, with or without cosmological constant, whose event horizons exhibit $\SU(2)$ symmetry must be that of a Berger sphere.

We investigate deformations of the Kerr-(A)dS near horizon geometry and derive partial infinitesimal rigidity results for it. The proof comprises two parts. First, we follow the analysis of Jezierski and Kami\'nski [Gen Rel Grav 45 (2013) 987--1004] to eliminate all but a finite number of Fourier modes of linear perturbations. In the second part, we give an argument using analyticity to prove that there are no odd Fourier modes., Comment: 20 pages

As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field $X$ is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of an extreme Myers-Perry black hole with equal angular momenta in two distinct planes), and on certain bundles over K\"ahler-Einstein manifolds. We find that these spaces exhibit a mild form of rigidity: they always admit a one-parameter group of isometries generated by $X$. Further geometrical and topological restrictions are also obtained.

Static vacuum near horizon geometries are solutions $(M,g,X)$ of a certain quasi-Einstein equation on a closed manifold $M$, where $g$ is a Riemannian metric and $X$ is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: $X$ vanishes and consequently $g$ is Ricci flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if $X$ does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and $(M,g)$ must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, $X$ must be exact (and vanishing if $\dim M=2$). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form $X$ to the closed $X$ case. We consider near horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An appendix contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary $X$)., Comment: Version accepted for publication

We prove a Bakry-\'Emery generalization of a theorem of Petersen and Wilhelm, itself a generalization of a theorem of Frankel, that closed minimal hypersurfaces in a complete manifold with a suitable curvature bound must intersect. We then prove splitting theorems of Croke-Kleiner type for manifolds bounded by hypersurfaces obeying Bakry-\'Emery curvature bounds. Motivated in part by the near-horizon geometry programme of general relativity, we do not assume that the Bakry-\'Emery vector field is of gradient type., Comment: New citations incorporated

We find and classify self-similar solutions of the curve shortening flow in standard hyperbolic 2-space. Together with earlier work of Halld\'orsson on curve shortening flow in the plane and Santos dos Reis and Tenenblat in the 2-sphere, this completes the classification of self-similar curve shortening flows in the constant curvature model spaces in 2-dimensions.

Is the universe finite or infinite, and what shape does it have? These fundamental questions, of which relatively little is known, are typically studied within the context of the standard model of cosmology where the universe is assumed to be homogeneous and isotropic. Here we address the above questions in highly general cosmological models, with the only assumption being that the average flow of matter is irrotational. Using techniques from differential geometry, specifically extensions of the Bonnet-Myers theorem, we derive a condition which implies a finite universe and yields a bound for its diameter. Furthermore, under a weaker condition involving the interplay between curvature and diameter, together with the assumption that the universe is finite (i.e., has closed spatial slices), we provide a concise list of possible topologies. Namely, the spatial sections then would be either the ring topologies $S^1 \times S^2$, $S^1\tilde{\times}S^2$, $S^1\times\mathbb{RP}^2$, $\mathbb{RP}^3 \# \mathbb{RP}^3$, or covered by the sphere $S^3$ or torus $T^3$. In particular, under this condition the basic construction of connected sums would be ruled out (save for one), along with the plethora of topologies associated with negative curvature. These results are obtained from consequences of the geometrization of 3-manifolds, by applying a generalization of the almost splitting theorem together with a curvature formula of Ehlers and Ellis., Comment: 12 pages; final version

The almost splitting theorem of Cheeger-Colding is established in the setting of almost nonnegative generalized $m$-Bakry-\'{E}mery Ricci curvature, in which $m$ is positive and the associated vector field is not necessarily required to be the gradient of a function. In this context it is shown that with a diameter upper bound and volume lower bound the fundamental group of such manifolds is almost abelian. Furthermore, extensions of well-known results concerning Ricci curvature lower bounds are given for generalized $m$-Bakry-\'{E}mery Ricci curvature. These include: the first Betti number bound of Gromov and Gallot, Anderson's finiteness of fundamental group isomorphism types, volume comparison, the Abresch-Gromoll inequality, and a Cheng-Yau gradient estimate. Finally, this analysis is applied to stationary vacuum black holes in higher dimensions to find that low temperature horizons must have limited topology, similar to the restrictions exhibited by (extreme) horizons of zero temperature., Comment: Comm. Math. Phys., to appear

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

It has been observed by Maldacena that one can extract asymptotically anti-de Sitter Einstein $4$-metrics from Bach-flat spacetimes by imposing simple principles and data choices. We cast this problem in a conformally compact Riemannian setting. Following an approach pioneered by Fefferman and Graham for the Einstein equation, we find formal power series for conformally compactifiable, asymptotically hyperbolic Bach-flat 4-metrics expanded about conformal infinity. We also consider Bach-flat metrics in the special case of constant scalar curvature and in the special case of constant $Q$-curvature. This allows us to determine the free data at conformal infinity, and to select those choices that lead to Einstein metrics. Interestingly, the mass is part of that free data, in contrast to the pure Einstein case. We then choose a convenient generalization of the Bach tensor to (bulk) dimensions $n>4$ and consider the higher dimensional problem. We find that the free data for the expansions split into low-order and high-order pairs. The former pair consists of the metric on the conformal boundary and its first radial derivative, while the latter pair consists of the radial derivatives of order $n-2$ and $n-1$. Higher dimensional generalizations of the Bach tensor lack some of the geometrical meaning of the 4-dimensional case. This is reflected in the relative complexity of the higher dimensional problem, but we are able to obtain a relatively complete result if conformal infinity is not scalar flat., Comment: Section 5 shortened and affiliations updated to match version accepted for publication

We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross-sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of $m$-quasi Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes., Comment: 11 pages; minor improvements

We show that in any spacetime dimension $D\ge 4$, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independent proofs are presented. One proof follows easily from differential geometry in the near-horizon limit, while others use Bakry-\'Emery-Ricci bounds for static Einstein manifolds., Comment: 8 pages; minor improvements

We show that near-horizon geometries in the presence of a positive cosmological constant cannot exist with ring topology. In particular, de Sitter black rings with vanishing surface gravity do not exist. Our result relies on a known mathematical theorem which is a straightforward consequence of a type of energy condition for a modified Ricci tensor, similar to the curvature-dimension conditions for the $m$-Bakry-\'Emery-Ricci tensor., Comment: 4 pages

We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-\'Emery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions $N\le 1$, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when $N=1$. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry-\'Emery-Ricci tensor for all $N\in (-\infty, 2]\cup (n,\infty)$ and for the $N=\infty$ case as well, with reduced rigidity if $N=2$. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-\'Emery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension., Comment: Very minor changes to match submitted version

With respect to any special boundary defining function, a conformally compact asymptotically hyperbolic metric has an asymptotic expansion near its conformal infinity. If this expansion is even to a certain order and satisfies one extra condition, then it is possible to define its renormalized volume and show that it is independent of choices that preserve this evenness structure. We prove that such expansions are preserved under normalized Ricci flow. We also study the variation of curvature functionals in this setting, and as one application, obtain the variation formula $$ \frac{{\rm d}}{{\rm d}t} {\rm RenV}\big(M^n, g(t)\big) = -\mathop{\vphantom{T}}^R \! \! \! \int_{M^n} (S(g(t))+n(n-1)) {\rm d}V_{g(t)},$$ where $S(g(t))$ is the scalar curvature for the evolving metric $g(t)$, and $\mathop{\vphantom{T}}^R \! \! \! \int (\cdot) {\rm d}V_g$ is Riesz renormalization. This extends our earlier work to a broader class of metrics., Comment: 21 pages

The "new positive energy conjecture" Horowitz and Myers (1999) probes a possible nonsupersymmetric AdS/CFT correspondence. We consider a version formulated for complete, asymptotically Poincar\'e-Einstein Riemannian metrics $(M,g)$ with bounded scalar curvature $R\ge -n(n-1)$. This version then asserts that any such $(M,g)$ must have mass not less than the mass $m_0$ of a metric $g_0$ induced on a time-symmetric slice of a certain AdS soliton spacetime. The conjecture remains unproved, having so far resisted standard techniques. Little is known other than that the conjecture is true for metrics which are sufficiently small perturbations of $g_0$. We pose another test for the conjecture. We assume its validity and attempt to prove as a corollary the corresponding scalar curvature rigidity statement, that $g_0$ is the unique asymptotically Poincar\'e-Einstein metric with mass $m=m_0$ obeying $R\ge -n(n-1)$. Were a second such metric $g_1$ not isometric to $g_0$ to exist, it then may well admit perturbations of lower mass, contradicting the assumed validity of the conjecture. We find that the minimum mass metric must be static Einstein, so the problem is reduced to that of static uniqueness. When $n=3$ the manifold is isometric to a time-symmetric slice of an AdS soliton spacetime, unless it has a non-compact horizon. En route we study the mass aspect, obtaining and generalizing known results. The mass aspect is (i) related to the holographic energy density, (ii) a weighted invariant under boundary conformal transformations when the bulk dimension is odd, and (iii) zero for negative Einstein manifolds with Einstein conformal boundary., Comment: Statement and proof of Lemma 3.1 corrected, other minor changes

We study Lorentzian manifolds with a weight function such that the $N$-Bakry-\'Emery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-\'Emery" $N= \infty$ case with $f$ uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite $N$-values $N\in (n,\infty)$ and $N\in (-\infty,1]$. In the $N\in (n,\infty)$ case, no bound on $f$ is required, while for $N\in (-\infty,1]$ and $N= \infty$, we are able to replace the boundedness of $f$ by a weaker condition on the integral of $f$ along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when $N=1$ the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in prior work on the $N$-Bakry-\'Emery curvature in Riemannian signature when $N=1$, and appears to be a general feature., Comment: In this version, a completeness assumption in theorems 1.9, 1.10, and 1.11 that had been too strong is now stated correctly

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initial metrics which belong to this class and are rotationally symmetric, we prove that if the sectional curvature in planes tangent to the orbits of symmetry is initially nonpositive, the flow starting from such an initial metric exists for all time. Moreover, if the sectional curvature in planes tangent to these orbits is initially negative, the flow converges at an exponential rate to standard hyperbolic space. This restriction on sectional curvature automatically rules out initial data admitting a minimal hypersphere., Comment: 28 pages, replaced one-word error in two locations on page 4

The classification of solutions of the static vacuum Einstein equations, on a given closed manifold or an asymptotically flat one, is a long-standing and much-studied problem. Solutions are characterized by a complete Riemannian $n$-manifold $(M,g)$ and a positive function $N$, called the lapse. We study this problem on Asymptotically Poincar\'e-Einstein $n$-manifolds, $n\ge 3$, when the conformal boundary-at-infinity is either a round sphere, a flat torus or smooth quotient thereof, or a compact hyperbolic manifold. Such manifolds have well-defined Wang mass, and are time-symmetric slices of static, vacuum, asymptotically anti-de Sitter spacetimes. By integrating a mildly generalized form of an identity used by Lindblom, Shen, Wang, and others, we give a mass formula for such manifolds. In consequence, we observe that either the lapse is trivial and $(M,g)$ is Poincar\'e-Einstein or the Wang mass is negative, as in the case of time symmetric slices of the AdS soliton. As an application, we use the mass formula to compute the renormalized volume of the warped product $(X,\gamma) = (M^3,g) \times_{N^2} (S^1,dt^2)$. We also give a mass formula for the case of a metric that is static in the region exterior to a horizon on which the lapse function is zero. Then the manifold $(X,\gamma)$ is said to have a "bolt" where the $S^1$ factor shrinks to zero length. The renormalized volume of $(X,\gamma)$ is expected on physical grounds to have the form of the free energy per unit temperature for a black hole in equilibrium with a radiation bath at fixed temperature. When $M$ is 3-dimensional and admits a horizon, we apply this mass formula to compute the renormalized volume of $(X,\gamma)$ and show that it indeed has the expected thermodynamically motivated form., Comment: Trivial correction to equation 2.1 (published version does not contain this correction)

We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-\'Emery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-\'Emery-Ricci tensor and obtain singularity theorems of a cosmological type, both for zero and for positive cosmological constant. That is, we find conditions under which every timelike geodesic is incomplete. These conditions are given by "open" inequalities, so we examine the borderline (equality) cases and show that certain singularities are avoided in these cases only if the geometry is rigid; i.e., if it splits as a Lorentzian product or, for a positive cosmological constant, a warped product, and the weight function is constant along the time direction. Then the product case is future timelike geodesically complete while, in the warped product case, worldlines of certain conformally static observers are complete. Our results answer a question posed by J Case. We then apply our results to the cosmology of scalar-tensor gravitation theories. We focus on the Brans-Dicke family of theories in 4 spacetime dimensions, where we obtain "Jordan frame" singularity theorems for big bang singularities., Comment: 15 pages; The wording of Theorem 1.5 is slightly clarified over the wording in the published version, with no change in the result

We study the evolution of the renormalized volume functional for asymptotically Poincare-Einstein metrics (M,g) which are evolving by normalized Ricci flow. In particular, we prove that the time derivative of the renormalized volume along the flow is the negative integral of scal(g(t)) + n(n-1) over the manifold. This implies that if scal(g(0))+n(n-1) is non-negative at t=0, then the renormalized volume decreases monotonically. We also discuss how, when n=4, our results describe the Hawking-Page phase transition. Differences in renormalized volumes give rigorous meaning to the Hawking-Page difference of actions and describe the free energy liberated in the transition., Comment: 15 pages; errors corrected and section 4 modified

For n+1 dimensional asymptotically AdS spacetimes which have holographic duals on their n dimensional conformal boundaries, we show that the imposition of causality on the boundary theory is sufficient to prove positivity of mass for the spacetime when n > 2, without the assumption of any local energy condition. We make crucial use of a generalization of the time-delay formula calculated in gr-qc/9404019, which relates the time delay of a bulk null curve with respect to a boundary null geodesic to the Ashtekar-Magnon mass of the spacetime. We also discuss holographic causality for the negative mass AdS soliton and its implications for the positive energy conjecture of Horowitz and Myers., Comment: Latex, 10 pages, 1 figure. Statement of theorem and proof modified. Typos corrected