Publications of SPP2026
On this site you find preprints and publications produced within the projects and with the support of the DFG priority programme „Geometry at Infinity“.
In this paper we study n-dimensional Ricci flows (M,g(t)), t in [0,T), where T is finite, and potentially a singular time, and for which the spatial L^p norm, p>n/2, of the scalar curvature is uniformly bounded on [0,T).
In the case that M is closed, we show that non-collapsing and non-inflating estimates hold. If we further assume that n=4 or that M^n is Kähler, we explain how these non-inflating/non-collapsing estimates can be combined with integral bounds on the Ricci and full curvature tensor of the prequel paper to obtain an improved space time integral bound of the Ricci curvature.
As an application of these estimates, we show that if we further restrict to n=4, then the solution convergences to an orbifold as t approaches T and that the flow can be extended using the Orbifold Ricci flow to the time interval [0,T+a)$ for some a>0.
We also prove local versions of many of the results mentioned above.
Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed n-dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial L^p sense for some p>2, then the estimates imply a uniform bound on the spatial L^2 norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition. In a sequel paper, we show that in many natural settings, a non-inflating condition holds.
Related project(s):
75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
We extend the K-cowaist inequality to generalized Dirac operators in the sense of Gromov and Lawson and study applications to manifolds with boundary.
| Journal | Comptes Rendus Mathématique |
| Publisher | Académie des Sciences, Institut de France |
| Volume | 362 |
| Pages | 1349-1356 |
| Link to preprint version | |
| Link to published version |
Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds52Spaces and Moduli Spaces of Riemannian Metrics with Curvature Bounds on compact and non-compact Manifolds II
In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by Müller zum Hagen--Robinson--Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n+1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani--Mazzieri and by Nozawa--Shiromizu--Izumi--Yamada.
Related project(s):
41Geometrically defined asymptotic coordinates in general relativity
It is a well-known fact that the Schwarzschild spacetime admits a maximal spacetime extension in null coordinates which extends the exterior Schwarzschild region past the Killing horizon, called the Kruskal-Szekeres extension. This method of extending the Schwarzschild spacetime was later generalized by Brill-Hayward to a class of spacetimes of "profile h" across non-degenerate Killing horizons. Circumventing analytical subtleties in their approach, we reconfirm this fact by reformulating the problem as an ODE, and showing that the ODE admits a solution if and only if the naturally arising Killing horizon is non-degenerate. Notably, this approach lends itself to discussing regularity across the horizon for non-smooth metrics.
We will discuss applications to the study of photon surfaces, extending results by Cederbaum-Galloway and Cederbaum-Jahns-Vičánek-Martínez beyond the Killing horizon. In particular, our analysis asserts that photon surfaces approaching the Killing horizon must necessarily cross it.
| Journal | Letters in Mathematical Physics |
| Publisher | Springer |
| Link to preprint version | |
| Link to published version |
Related project(s):
41Geometrically defined asymptotic coordinates in general relativity
We present a new proof of the Willmore inequality for an arbitrary bounded domain Ω⊂ℝ^n with smooth boundary. Our proof is based on a parametric geometric inequality involving the electrostatic potential for the domain Ω; this geometric inequality is derived from a geometric differential inequality in divergence form. Our parametric geometric inequality also allows us to give new proofs of the quantitative Willmore-type and the weighted Minkowski inequalities by Agostiniani and Mazzieri.
Related project(s):
41Geometrically defined asymptotic coordinates in general relativity
We present different proofs of the uniqueness of 4-dimensional static vacuum asymptotically flat spacetimes containing a connected equipotential photon surface or in particular a connected photon sphere. We do not assume that the equipotential photon surface is outward directed or non-degenerate and hence cover not only the positive but also the negative and the zero mass case which has not yet been treated in the literature. Our results partially reproduce and extend beyond results by Cederbaum and by Cederbaum and Galloway. In the positive and negative mass cases, we give three proofs which are based on the approaches to proving black hole uniqueness by Israel, Robinson, and Agostiniani--Mazzieri, respectively. In the zero mass case, we give four proofs. One is based on the positive mass theorem, the second one is inspired by Israel's approach and in particular leads to a new proof of the Willmore inequality in (ℝ^3,δ), under a technical assumption. The remaining two proofs are inspired by proofs of the Willmore inequality by Cederbaum and Miehe and by Agostiniani and Mazzieri, respectively. In particular, this suggests to view the Willmore inequality and its rigidity case as a zero mass version of equipotential photon surface uniqueness.
Related project(s):
41Geometrically defined asymptotic coordinates in general relativity
We study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restrictionon the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.
| Journal | Annales Henri Poincaré |
| Publisher | Springer |
| Link to preprint version | |
| Link to published version |
Related project(s):
41Geometrically defined asymptotic coordinates in general relativity
We extend the classical theory of homotopical Σ-sets, defined by Bieri, Neumann, Renz and Strebel for abstract groups, to locally compact Hausdorff groups. Given such a group G, our geometric invariants are sets of continuous homomorphisms G→R ("characters"). They match the classical Σ-sets if G is discrete, and refine the homotopical compactness properties of Abels and Tiemeyer. Moreover, our theory recovers the definition of low-dimensional geometric invariants for topological gropus proposed by Kochloukova.
Related project(s):
39Geometric invariants of discrete and locally compact groups
We prove that the group of isometries preserving a metric foliation on a closed Alexandrov space \(X\), or a singular Riemannian foliation on a manifold \(M\) is a closed subgroup of the isometry group of \(X\) in the case of a metric foliation, or of the isometry group of \(M\) for the case of a singular Riemannian foliation. We obtain a sharp upper bound for the dimension of these subgroups and show that, when equality holds, the foliations that realize this upper bound are induced by fiber bundles whose fibers are round spheres or projective spaces. Moreover, singular Riemannian foliations that realize the upper bound are induced by smooth fiber bundles whose fibers are round spheres or projective spaces.
Related project(s):
43Singular Riemannian foliations and collapse
In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang [14]. By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in [14]. More precisely, we prove that if the graph induced by a closed $1$-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.
| Journal | Annals of PDE |
| Link to preprint version | |
| Link to published version |
Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in \(\mathbb R^{n+1}\), and prove the existence and regularity of the flow before extincting to a point in finite time.
| Journal | Advanced Nonlinear Studies |
| Link to preprint version | |
| Link to published version |
Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
In this paper, we show the relation between the existence of twisted conical Kähler-Ricci solitons and the greatest log Bakry-Emery-Ricci lower bound on Fano manifolds. This is based on our proofs of some openness theorems on the existence of twisted conical Kähler-Ricci solitons, which generalize Donaldson's existence conjecture and openness theorem of the conical Kähler-Einstein metrics to the conical soliton case.
| Journal | Science China Mathematics |
| Link to preprint version | |
| Link to published version |
Related project(s):
31Solutions to Ricci flow whose scalar curvature is bounded in Lp.75Solutions to Ricci flow whose scalar curvature is bounded in L^p II
For vector-valued Maass cusp forms for SL(2,Z) with real weight k∈R and spectral parameter s∈C, Res∈(0,1), s≢±k/2 mod 1, we propose a notion of vector-valued period functions, and we establish a linear isomorphism between the spaces of Maass cusp forms and period functions by means of a cohomological approach. The period functions are a generalization of those for the classical Maass cusp forms, being solutions of a finite-term functional equation or, equivalently, eigenfunctions with eigenvalue 1 of a transfer operator deduced from the geodesic flow on the modular surface. We apply this result to deduce a notion of period functions and related linear isomorphism for Jacobi Maass forms of weight k+1/2 for the semi-direct product of SL2(Z) with the integer points Hei(Z) of the Heisenberg group.
Related project(s):
70Spectral theory with non-unitary twists
For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for Kähler manifolds, Kähler-Einstein manifolds, quaternionic Kähler manifolds and manifolds with a harmonic 1-form of constant length.
Related project(s):
37Boundary value problems and index theory on Riemannian and Lorentzian manifolds
We investigate groups that act amenably on their Higson corona (also known as bi-exact groups) and we provide reformulations of this in relation to the stable Higson corona, nuclearity of crossed products and to positive type kernels. We further investigate implications of this in relation to the Baum-Connes conjecture, and prove that Gromov hyperbolic groups have isomorphic equivariant K-theories of their Gromov boundary and their stable Higson corona.
Related project(s):
45Macroscopic invariants of manifolds
We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first ℓ2-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.
Related project(s):
58Profinite perspectives on l2-cohomology
We prove that every finitely generated, residually finite group G embeds into a finitely generated perfect branch group such that many properties of G are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non-abelian free group. As an application, we construct a finitely generated, non-amenable torsion branch group.
| Journal | Bull. Lond. Math. Soc. |
| Volume | 56 |
| Link to preprint version | |
| Link to published version |
Related project(s):
58Profinite perspectives on l2-cohomology
We study \(\mathsf{RCD}\)-spaces \((X,d,\mathfrak{m})\) with group actions by isometries preserving the reference measure \(\mathfrak{m}\) and whose orbit space has dimension one, i.e. cohomogeneity one actions. To this end we prove a Slice Theorem asserting that each slice at a point is homeomorphic to a non-negatively curved \(\mathsf{RCD}\)-space. Under the assumption that \(X\) is non-collapsed we further show that the slices are homeomorphic to metric cones over homogeneous spaces with \(\mathrm{Ric}\geq 0\). As a consequence we obtain complete topological structural results and a principal orbit representation theorem. Conversely, we show how to construct new \(\mathsf{RCD}\)-spaces from a cohomogeneity one group diagram, giving a complete description of \(\mathsf{RCD}\)-spaces of cohomogeneity one. As an application of these results we obtain the classification of cohomogeneity one, non-collapsed \(\mathsf{RCD}\)-spaces of essential dimension at most 4.
Related project(s):
43Singular Riemannian foliations and collapse
We show that the sign of the Euler characteristic of an S-arithmetic subgroup of a simple k-group over a number field k depends on the S-congruence completion only. Consequently, the sign is a profinite invariant for such S-arithmetic groups with the congruence subgroup property. This generalizes previous work of the first author with Kionke-Raimbault-Sauer.
Related project(s):
58Profinite perspectives on l2-cohomology