arXiv subjects
Publications and source records attributed to Sven Hirsch.
We characterize all maximally charged black hole spacetimes in $3+1$ dimensions. More precisely, given any initial data set $(M,g,k)$ saturating the mass-charge inequality that satisfies the charged dominant energy condition, we show that $(M,g,k)$ must arise from an isometric embedding into a Majumdar-Papapetrou spacetime.
High-resolution 3D segmentation of hip and shoulder anatomy from CT and MRI is essential for surgical planning, yet frozen segmentation models often fail under domain shift. CNN-based expert models are fully automatic but lack adaptability, whereas promptable foundation models generalize better but require manual prompting. We present MedSAM2-Anatomy, a training-free inference-time optimization framework that improves frozen segmentation models without retraining or human interaction. A frozen expert model generates anatomical priors that are automatically converted into multiple prompt hypotheses for a frozen 3D foundation model. Candidate masks are fused while anatomically implausible priors are rejected. No model weights are updated and no manual prompts are required. TotalSegmentator and MedSAM2 are used as representative expert and foundation models, allowing the contribution of the inference policy to be isolated. Evaluation on the independent Balgrist-V0 CT and MRI cohorts shows that inference-time optimization increases median Dice from 0.71 to 0.92 on hip MRI and from 0.89 to 0.92 on shoulder CT, while reducing median HD95 on hip MRI from 22.0 mm to 5.0 mm. On public TotalSegmentator benchmarks, the expert model remains strongest, indicating that the optimal fusion strategy depends on the reliability of the expert prior. These results demonstrate that training-free inference-time optimization provides a practical strategy for improving frozen segmentation models without manual prompting.
We prove a sharp stable $2$-systolic inequality for complex projective space under the scalar curvature lower bound of the normalized Fubini-Study metric. If $M$ is diffeomorphic to $\mathbb{C}\mathrm{P}^n$ and $\mathrm{scal}_g\ge 4n(n+1)$, then $\mathrm{sys}_2^{\mathrm{st}}(M,g)\le \pi$. Moreover, equality holds only for the Fubini-Study metric, up to biholomorphism after choosing the corresponding complex structure. The proof uses Spin$^c$ Dirac operators, a comass estimate for the curvature term in the Lichnerowicz formula, and stable norm-comass duality.
We give a counterexample to a recently conjectured variant of the Penrose inequality.
Using the recent work of Brendle--Wang on the Riemannian positive mass theorem, we prove the spacetime positive mass theorem for asymptotically flat and asymptotically hyperboloidal initial data sets in arbitrary dimensions.
We generalize the spacetime positive mass theorem to include multiple time dimensions. In particular, we show that the mass remains nonnegative in the sense that the energy $E$ is bounded from below by the trace norm of the linear momenta $J^1,...,J^m$. Equality in this energy inequality implies a foliation by flat submanifolds of a generalized initial data set. Moreover, under an additional umbilicity assumption, we find that the initial data set isometrically embeds into a generalized pp-wave.
We prove a Riemannian positive mass theorem for asymptotically flat spin manifolds with hypersurface singularities. Unlike earlier results, some components of the singular set may be mean-concave, provided that other components of the singular set are sufficiently mean-convex. Our proof uses initial data sets where a suitably chosen second fundamental form transfers convexity defects between different singularity components.
We prove Gromov's conjecture on the total mean curvature of fill-ins in various cases. Our methods are based on surgery to reduce the statement to fill-ins of spheres, which can be treated by instances of the positive mass theorem. For spin fill-ins, where we permit the mean curvature to take negative values, we build on a classical surgery result of Lawson-Michelsohn and a recent positive mass theorem with creases by Kazaras-Khuri-Lin. For non-spin fill-ins of spin manifolds, where we assume the mean curvature to be non-negative, we develop a novel quantitative surgery process to reduce the general situation to a result of Shi-Wang-Wei. We also treat the case of fill-ins of non-spin manifolds, provided there is a fixed positive lower bound on the mean curvature.
We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schr\"odinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity.
We characterize spin initial data sets that saturate the BPS bound in the asymptotically AdS setting. This includes both gravitational waves and rotating black holes in higher dimensions, and we establish a sharp dimension threshold in each case. A key ingredient in our argument is a theorem providing a general criterion for when an imaginary Killing spinor of mixed causal type can be replaced by one that is strictly timelike or null. Moreover, in analogy with the minimal surface method, we demonstrate that spinors can be used to construct a codimension-$2$ slicing.
The Circle of Willis (CoW) is a critical network of arteries in the brain, often implicated in cerebrovascular pathologies. Voxel-level segmentation is an important first step toward an automated CoW assessment, but a full quantitative analysis requires centerline representations. However, conventional skeletonization techniques often struggle to extract reliable centerlines due to the CoW's complex geometry, and publicly available centerline datasets remain scarce. To address these challenges, we used a thinning-based skeletonization algorithm to extract and curate centerline graphs and morphometric features from the TopCoW dataset, which includes 200 stroke patients, each imaged with MRA and CTA. The curated graphs were used to develop a baseline algorithm for centerline and feature extraction, combining U-Net-based skeletonization with A* graph connection. Performance was evaluated on a held-out test set, focusing on anatomical accuracy and feature robustness. Further, we used the extracted features to predict the frequency of fetal PCA variants, confirm theoretical bifurcation optimality relations, and detect subtle modality differences. The baseline algorithm consistently reconstructed graph topology with high accuracy (F1 = 1), and the average Euclidean node distance between reference and predicted graphs was below one voxel. Features such as segment radius, length, and bifurcation ratios showed strong robustness, with median relative errors below 5% and Pearson correlations above 0.95. Our results demonstrate the utility of learning-based skeletonization combined with graph connection for anatomically plausible centerline extraction. We emphasize the importance of going beyond simple voxel-based measures by evaluating anatomical accuracy and feature robustness. The dataset and baseline algorithm have been released to support further method development and clinical research.
We address two problems concerning the ADM mass-minimizing initial data sets. First, we show that the equality case of the positive mass theorem embeds into a pp-wave spacetime. Second, we show that positive Bartnik mass minimizers embed into strongly stationary vacuum spacetimes, thereby confirming the Bartnik stationary vacuum conjecture. A key ingredient is a new monotonicity formula for the Lorentzian length of a causal Killing vector field, which, among other applications, yields a strong maximum principle for the length.
In Special Relativity, massless objects are characterized as either vacuum states or as radiation propagating at the speed of light. This distinction extends to General Relativity for asymptotically flat initial data sets (IDS) \((M^n, g, k)\), where vacuum is represented by slices of Minkowski space, and radiation is modeled by slices of \(pp\)-wave spacetimes. In contrast, we demonstrate that asymptotically hyperboloidal IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations modeling radiation in this setting. Our result holds under the most general assumptions. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions.
We establish new mean curvature rigidity theorems for spin fill-ins with non-negative scalar curvature using two different spinorial techniques. Our results address two questions by Miao and Gromov, respectively. The first technique is based on extending boundary spinors satisfying a generalized eigenvalue equation via the Fredholm alternative for an APS boundary value problem, while the second is a comparison result in the spirit of Llarull and Lott using index theory. We also show that the latter implies a new Witten-type integral inequality for the mass of an asymptotically Schwarzschild manifold, which holds even when the scalar curvature is not assumed to be non-negative.
In 1981, Schoen-Yau and Witten showed that in General Relativity both the total energy $E$ and the total mass $m$ of an initial data set modeling an isolated gravitational system are non-negative. Moreover, if $E=0$, the initial data set must be contained in Minkowski space. In this paper, we show that if $m=0$, i.e. if $E$ equals the total momentum $|P|$, the initial data set must be contained in a pp-wave spacetime. Our proof combines spinorial methods with spacetime harmonic functions and works in all dimensions. Additionally, we find the decay rate threshold where the embedding has to be within Minkowski space and construct non-vacuum initial data sets with $m=0$ in the borderline case. As a consequence, this completely settles the rigidity of the spacetime positive mass theorem for spin manifolds.
Llarull's theorem characterizes the round sphere $S^n$ among all spin manifolds whose scalar curvature is bounded from below by $n(n-1)$. In this paper we show that if the scalar curvature is bounded from below by $n(n-1)-\varepsilon$, the underlying manifold is $C^0$-close to a finite number of spheres outside a small bad set. This completely solves Gromov's spherical stability problem.
We prove uniqueness of tangent cones for forced mean curvature flow, at both closed self-shrinkers and round cylindrical self-shrinkers, in any codimension. The corresponding results for mean curvature flow in Euclidean space were proven by Schulze and Colding-Minicozzi respectively. We adapt their methods to handle the presence of the forcing term, which vanishes in the blow-up limit but complicates the analysis along the rescaled flow. Our results naturally include the case of mean curvature flows in Riemannian manifolds.
The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not admit a metric of positive scalar curvature. In this work we introduce a new notion of curvature interpolating between Ricci and scalar curvature (so called $m$-intermediate curvature), and use stable weighted slicings to show that for $n \leq 7$ the manifolds $N^n = M^{n-m} \times \mathbb{T}^m$ do not admit a metric of positive $m$-intermediate curvature.