Differential Geometry

New submissions

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New submissions for Fri, 17 May 24

[1]  arXiv:2405.09604 [pdf, other]

Title: On the existence of geodesic vector fields on closed surfaces

Authors: Vladimir S. Matveev

Comments: 4 pages, 1 figure

Subjects: Differential Geometry (math.DG)

We construct an example of a Riemannian metric on the 2-torus such that its universal cover does not admit global Riemann normal coordinates.

[2]  arXiv:2405.09750 [pdf, ps, other]

Title: Scalar curvature lower bounds on asymptotically flat manifolds

Authors: Yuqiao Li

Subjects: Differential Geometry (math.DG)

In this paper, we consider the scalar curvature in the distributional sense of \cite{MR3366052} and the scalar curvature lower bound in the $\beta-$weak $(\beta\in(0, \frac{1}{2}))$ sense of \cite{MR4685089} on an asymptotically flat $n-$manifold with a $W^{1,p}(p>n)$ metric. We first show that the scalar curvature lower bound under the Ricci-DeTurck flow depends on the scalar curvature lower bound in the $\beta-$weak sense and the time. Then we prove that the lower bound of the distributional scalar curvature of a $W^{1, p}$ metric coincides with the lower bound of the scalar curvature in the $\beta-$weak sense at infinity.

[3]  arXiv:2405.09872 [pdf, ps, other]

Title: Conformal metrics with finite total Q-curvature revisited

Authors: Mingxiang Li

Comments: 21 pages

Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)

Given a conformal metric with finite total Q-curvature on $\mathbb{R}^n$ for $n\geq4$, we show that the sign of scalar curvature near infinity control not only the upper bound but also the lower bound of Q-curvature integral which is a new phenomenon. Meanwhile, for general complete non-compact four-manifolds with simple ends, we also obtain similar control of the lower bound of Q-curvature integral.

[4]  arXiv:2405.09969 [pdf, ps, other]

Title: The van Est homomorphism for strict Lie 2-groups

Authors: Camilo Angulo, Miquel Cueca

Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Symplectic Geometry (math.SG)

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.

Cross-lists for Fri, 17 May 24

[5]  arXiv:2405.09800 (cross-list from cs.LG) [pdf, other]

Title: Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution

Authors: Eslam Zaher, Maciej Trzaskowski, Quan Nguyen, Fred Roosta

Comments: Accepted at ICML 2024

Subjects: Machine Learning (cs.LG); Human-Computer Interaction (cs.HC); Differential Geometry (math.DG)

In this paper, we dive into the reliability concerns of Integrated Gradients (IG), a prevalent feature attribution method for black-box deep learning models. We particularly address two predominant challenges associated with IG: the generation of noisy feature visualizations for vision models and the vulnerability to adversarial attributional attacks. Our approach involves an adaptation of path-based feature attribution, aligning the path of attribution more closely to the intrinsic geometry of the data manifold. Our experiments utilise deep generative models applied to several real-world image datasets. They demonstrate that IG along the geodesics conforms to the curved geometry of the Riemannian data manifold, generating more perceptually intuitive explanations and, subsequently, substantially increasing robustness to targeted attributional attacks.

[6]  arXiv:2405.10003 (cross-list from math.AG) [pdf, ps, other]

Title: Bogomolov-Gieseker inequality for log terminal Kähler threefolds

Authors: Henri Guenancia, Mihai Păun

Comments: 37 pages

Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Differential Geometry (math.DG)

In this article we are concerned with two main themes revolving around compact K\"ahler spaces with log terminal singularities. The first one is the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves on threefolds while the second one has to do with the construction of canonical metrics on such spaces and their properties.

[7]  arXiv:2405.10065 (cross-list from math.GT) [pdf, other]

Title: Arc coordinates for maximal representations

Authors: Marta Magnani

Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

We generalize arc coordinates for maximal representations from a hyperbolic surface with boundary into $\text{PSp}(4,\mathbb{R})$, focusing on the case where the surface is a pair of pants. We introduce geometric parameters within the space of right-angled hexagons in the Siegel space $\mathcal{X}$. These parameters enable the visualization of a right-angled hexagon as a polygonal chain inside the hyperbolic plane $\mathbb{H}^{2}$. We explore the geometric properties of reflections in $\mathcal{X}$ and introduce the notion of maximal representation of the reflection group $W_{3}=\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}*\mathbb{Z}/2\mathbb{Z}$. We parametrize maximal representations from $W_{3}$ into $\text{PSp}^{\pm}(4,\mathbb{R})$, this induces a natural parametrization of a subset of maximal and Shilov hyperbolic representations into $\text{PSp}(4,\mathbb{R})$.

Replacements for Fri, 17 May 24

[8]  arXiv:1210.8070 (replaced) [pdf, ps, other]

Title: The eta invariant on two-step nilmanifolds

Authors: Ruth Gornet, Ken Richardson

Comments: 55 pages in this new version; corrected sign error found by M. Stone in computation of 3-dimensional Heisenberg Dirac eigenvalues

Journal-ref: Comm. Anal. Geom. 26 (2018), no. 2, 271-346

Subjects: Differential Geometry (math.DG); K-Theory and Homology (math.KT); Spectral Theory (math.SP)

[9]  arXiv:1803.01615 (replaced) [pdf, ps, other]

Title: On the index of minimal 2-tori in the 4-sphere

Authors: Rob Kusner, Peng Wang

Comments: 12 pages

Journal-ref: Journal f\"ur die reine und angewandte Mathematik (Crelles Journal),vol. 2024, no. 806, 2024, pp. 9-22

Subjects: Differential Geometry (math.DG)

[10]  arXiv:2211.06916 (replaced) [pdf, other]

Title: On spectral simplicity of the Hodge Laplacian and Curl Operator along paths of metrics

Authors: Willi Kepplinger

Comments: final version, to appear in TAMS. The most important changes include a change in title (shortened), a more detailed proof of Theorem 1.4 and Lemma 3.3 as well as the inclusion of two new Lemmas (3.5 and 3.6) in section 3.2. Furthermore the Beltrami operator has been renamed into Curl operator at the suggestion of an anonymous referee

Subjects: Differential Geometry (math.DG); Spectral Theory (math.SP)

[11]  arXiv:2212.13808 (replaced) [pdf, ps, other]

Title: Index estimates for sequences of harmonic maps

Authors: Jonas Hirsch, Tobias Lamm

Comments: minor modifications, to appear in Comm. Anal. Geom

Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)

[12]  arXiv:2303.15333 (replaced) [pdf, ps, other]

Title: A note on the long neck principle and spectral width inequality of geodesic collar neighborhoods

Authors: Daoqiang Liu

Comments: Final version, to appear in the Proceedings of the American Mathematical Society

Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph)

[13]  arXiv:2310.09072 (replaced) [pdf, ps, other]

Title: Conformal Kaehler submanifolds

Authors: L. J. Alías, S. Chion, M. Dajczer

Subjects: Differential Geometry (math.DG)

[14]  arXiv:2404.09449 (replaced) [pdf, ps, other]

Title: Scattering rigidity for standard stationary manifolds via timelike geodesics

Authors: Sebastián Muñoz-Thon

Comments: 26 pages; typos corrected, more explicit result and hypothesis, references added

Subjects: Differential Geometry (math.DG)

[15]  arXiv:2109.10704 (replaced) [pdf, other]

Title: Proof of the $C^2$-stability conjecture for geodesic flows of closed surfaces

Authors: Gonzalo Contreras, Marco Mazzucchelli

Comments: 34 pages, 6 figures; final version, as published

Journal-ref: Duke Mathematical Journal 173 (2024), no. 2, 347-390

Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Symplectic Geometry (math.SG)

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