arXiv subjects

Felix Schulze

Publications and source records attributed to Felix Schulze.

At least 37 records · Page 2Linked to original sources

arXiv2022

Neck pinches along the Lagrangian mean curvature flow of surfaces

Let $L_t$ be a zero Maslov, rational Lagrangian mean curvature flow in a compact Calabi-Yau surface, and suppose that at the first singular time a tangent flow is given by the static union of two transverse planes. We show that in this case the tangent flow is unique, and that the flow can be continued past the singularity as an immersed, smooth, zero Maslov, rational Lagrangian mean curvature flow. Furthermore, if $L_0$ is a sphere that is stable in the sense of Thomas-Yau, then such a singularity cannot form.

math.DG↗

arXiv2021

Galaxy Lookback Evolution Models -- a Comparison with Magneticum Cosmological Simulations and Observations

We construct empirical models of star-forming galaxy evolution assuming that individual galaxies evolve along well-known scaling relations between stellar mass, gas mass and star formation rate following a simple description of chemical evolution. We test these models by a comparison with observations and with detailed Magneticum high resolution hydrodynamic cosmological simulations. Galaxy star formation rates, stellar masses, gas masses, ages, interstellar medium and stellar metallicities are compared. It is found that these simple lookback models capture many of the crucial aspects of galaxy evolution reasonably well. Their key assumption of a redshift dependent power law relationship between galaxy interstellar medium gas mass and stellar mass is in agreement with the outcome of the complex Magneticum simulations. Star formation rates decline towards lower redshift not because galaxies are running out of gas, but because the fraction of the cold ISM gas, which is capable of producing stars, becomes significantly smaller. Gas accretion rates in both model approaches are of the same order of magnitude. Metallicity in the Magneticum simulations increases with the ratio of stellar mass to gas mass as predicted by the lookback models. The mass metallicity relationships agree and the star formation rate dependence of these relationships is also reproduced. We conclude that these simple models provide a powerful tool for constraining and interpreting more complex models based on cosmological simulations and for population synthesis studies analyzing integrated spectra of stellar populations.

astro-ph.GA↗

arXiv2021

The SAMI Galaxy Survey: a statistical approach to an optimal classification of stellar kinematics in galaxy surveys

Large galaxy samples from multi-object IFS surveys now allow for a statistical analysis of the z~0 galaxy population using resolved kinematics. However, the improvement in number statistics comes at a cost, with multi-object IFS survey more severely impacted by the effect of seeing and lower S/N. We present an analysis of ~1800 galaxies from the SAMI Galaxy Survey and investigate the spread and overlap in the kinematic distributions of the spin parameter proxy $λ_{Re}$ as a function of stellar mass and ellipticity. For SAMI data, the distributions of galaxies identified as regular and non-regular rotators with \textsc{kinemetry} show considerable overlap in the $λ_{Re}$-$\varepsilon_e$ diagram. In contrast, visually classified galaxies (obvious and non-obvious rotators) are better separated in $λ_{Re}$ space, with less overlap of both distributions. Then, we use a Bayesian mixture model to analyse the observed $λ_{Re}$-$\log(M_*/M_{\odot})$ distribution. Below $\log(M_{\star}/M_{\odot})\sim10.5$, a single beta distribution is sufficient to fit the complete $λ_{Re}$ distribution, whereas a second beta distribution is required above $\log(M_{\star}/M_{\odot})\sim10.5$ to account for a population of low-$λ_{Re}$ galaxies. While the Bayesian mixture model presents the cleanest separation of the two kinematic populations, we find the unique information provided by visual classification of kinematic maps should not be disregarded in future studies. Applied to mock-observations from different cosmological simulations, the mixture model also predicts bimodal $λ_{Re}$ distributions, albeit with different positions of the $λ_{Re}$ peaks. Our analysis validates the conclusions from previous smaller IFS surveys, but also demonstrates the importance of using kinematic selection criteria that are dictated by the quality of the observed or simulated data.

astro-ph.GA↗

arXiv2021

A relative entropy and a unique continuation result for Ricci expanders

We prove an optimal relative integral convergence rate for two expanding gradient Ricci solitons coming out of the same cone. As a consequence, we obtain a unique continuation result at infinity and we prove that a relative entropy for two such self-similar solutions to the Ricci flow is well-defined.

math.DG↗

arXiv2020

Uniqueness of asymptotically conical tangent flows

Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to "flow through" such a singularity.

math.DG↗

arXiv2020

Positive solutions to Schrödinger equations and geometric applications

A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

math.DG↗

arXiv2020

Remarks on the self-shrinking Clifford torus

On the one hand, we prove that the Clifford torus in $\mathbb{C}^2$ is unstable for Lagrangian mean curvature flow under arbitrarily small Hamiltonian perturbations, even though it is Hamiltonian $F$-stable and locally area minimising under Hamiltonian variations. On the other hand, we show that the Clifford torus is rigid: it is locally unique as a self-shrinker for mean curvature flow, despite having infinitesimal deformations which do not arise from rigid motions. The proofs rely on analysing higher order phenomena: specifically, showing that the Clifford torus is not a local entropy minimiser even under Hamiltonian variations, and demonstrating that infinitesimal deformations which do not generate rigid motions are genuinely obstructed.

math.DG↗

arXiv2020

On the regularity of Ricci flows coming out of metric spaces

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)), x_0) \to (M,d_0, x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff sense. In the case that $B_{g(t)}(x_0,1) \Subset M$ for all $t\in (0,T)$ and $d_0$ is generated by a smooth Riemannian metric in distance coordinates, we show using Ricci-harmonic map heat flow, that there is a corresponding smooth solution $\tilde g(t)_{t\in (0,T)}$ to the $δ$-Ricci-DeTurck flow on an Euclidean ball ${\mathbb B}_{r}(p_0) \subset {\mathbb R}^n$, which can be extended to a smooth solution defined for $t \in [0,T)$. We further show, that this implies that the original solution $g$ can be extended to a smooth solution on $B_{d_0}(x_0,r/2)$ for $t\in [0,T)$, in view of the method of Hamilton.

math.DG↗

arXiv2020

Ancient solutions in Lagrangian mean curvature flow

Ancient solutions of Lagrangian mean curvature flow in C^n naturally arise as Type II blow-ups. In this extended note we give structural and classification results for such ancient solutions in terms of their blow-down and, motivated by the Thomas-Yau Conjecture, focus on the almost calibrated case. In particular, we classify Type II blow-ups of almost calibrated Lagrangian mean curvature flow when the blow-down is a pair of transverse planes or, when n=2, a multiplicity two plane. We also show that the Harvey-Lawson Clifford torus cone in C^3 cannot arise as the blow-down of an almost calibrated Type II blow-up.

math.DG↗

arXiv2020

Generic uniqueness of expanders with vanishing relative entropy

We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that expanders with vanishing relative entropy are unique in a generic sense. This also implies that generically locally entropy minimising expanders are unique.

math.DG↗

arXiv2020

Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $Σ$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\partial S = Σ$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $Σ$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \sqrtπ\, \mathbf{M}[S] \leq (\mathbf{M}[Σ])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by $-κ< 0$ and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in $L^2$.

math.DG↗

arXiv2020

Kinematics of Simulated Galaxies II: Probing the Stellar Kinematics of Galaxies out to Large Radii

We investigate the stellar kinematics of a sample of galaxies extracted from the hydrodynamic cosmological Magneticum Pathfinder simulations out to $5$ half-mass radii. We construct differential radial stellar spin profiles quantified by the observationally widely used $λ_\mathrm{R}$ and the closely related $(V/σ)$ parameters. We find three characteristic profile shapes: profiles exhibiting a (i) peak within $2.5$ half-mass radii and a subsequent decrease (ii) continuous increase that plateaus at larger radii typically with a high amplitude (iii) completely flat behaviour typically with low amplitude, in agreement with observations. This shows that the kinematic state of the stellar component can vary significantly with radius, suggesting a distinct interplay between in-situ star formation and ex-situ accretion of stars. Following the evolution of our sample through time, we provide evidence that the accretion history of galaxies with decreasing profiles is dominated by the anisotropic accretion of low mass satellites that get disrupted beyond $ \sim 2.0$ half-mass radii, building up a stellar halo with non-ordered motion while maintaining the central rotation already present at $z=2$. In fact, at $z=2$ decreasing profiles are the predominant profile class. Hence, we can predict a distinct formation pathway for galaxies with a decreasing profile and show that the centre resembles an old embedded disk. Furthermore, we show that the radius of the kinematic transition provides a good estimation for the transition radius from in-situ stars in the centre to accreted stars in the halo.

astro-ph.GA↗

arXiv2019

The SAMI Galaxy Survey: comparing 3D spectroscopic observations with galaxies from cosmological hydrodynamical simulations

Cosmological hydrodynamical simulations are rich tools to understand the build-up of stellar mass and angular momentum in galaxies, but require some level of calibration to observations. We compare predictions at $z\sim0$ from the Eagle, Hydrangea, Horizon-AGN, and Magneticum simulations with integral field spectroscopic (IFS) data from the SAMI Galaxy Survey, ATLAS3D, CALIFA and MASSIVE surveys. The main goal of this work is to simultaneously compare structural, dynamical, and stellar population measurements in order to identify key areas of success and tension. We have taken great care to ensure that our simulated measurement methods match the observational methods as closely as possible. We find that the Eagle and Hydrangea simulations reproduce many galaxy relations but with some offsets at high stellar masses. There are moderate mismatches in $R_e$ (+), $ε$ (-), $σ_e$ (-), and mean stellar age (+), where a plus sign indicates that quantities are too high on average, and minus sign too low. The Horizon-AGN simulations qualitatively reproduce several galaxy relations, but there are a number of properties where we find a quantitative offset to observations. Massive galaxies are better matched to observations than galaxies at low and intermediate masses. Overall, we find mismatches in $R_e$ (+), $ε$ (-), $σ_e$ (-) and $(V/σ)_e$ (-). Magneticum matches observations well: this is the only simulation where we find ellipticities typical for disk galaxies, but there are moderate differences in $σ_e$ (-), $(V/σ)_e$ (-) and mean stellar age (+). Our comparison between simulations and observational data has highlighted several areas for improvement, such as the need for improved modelling resulting in a better vertical disk structure, yet our results demonstrate the vast improvement of cosmological simulations in recent years.

astro-ph.GA↗

arXiv2019

Local foliation of manifolds by surfaces of Willmore type

We show the existence of a local foliation of a three dimensional Riemannian manifold by critical points of the Willmore functional subject to a small area constraint around non-degenerate critical points of the scalar curvature. This adapts a method developed by Rugang Ye to construct foliations by surfaces of constant mean curvature.

math.DG↗

arXiv2018

Kinematics of Simulated Galaxies I: Connecting Dynamical and Morphological Properties of Early-Type Galaxies at Different Redshifts

State-of-the-art integral field surveys like $\mathrm{ATLAS^{3D}}$, SLUGGS, CALIFA, SAMI, and MaNGA provide large data sets of kinematical observations of early-type galaxies (ETGs), yielding constraints on the formation of ETGs. Using the cosmological hydrodynamical \textit{Magneticum Pathfinder} simulations, we investigate the paradigm of fast and slow rotating ETGs in a fully cosmological context. We show that the ETGs within the \textit{Magneticum} simulation are in remarkable agreement with the observations, revealing fast and slow rotators quantified by the angular momentum proxy $λ_{\mathrm{R}}$ and the flattening $ε$ with the observed prevalence. Taking full advantage of the three-dimensional data, we demonstrate that the dichotomy between fast and slow rotating galaxies gets enhanced, showing an upper and lower population separated by an underpopulated region in the edge-on $λ_{\mathrm{R}}$-$ε$ plane. Following the evolution of the $λ_{\mathrm{R}}$-$ε$ plane through cosmic time, we find that, while the upper population is already in place at $z=2$, the lower population gets statistically significant below $z=1$ with a gradual increase. At least $50\%$ of the galaxies transition from fast to slow rotators on a short timescale, in most cases associated to a significant merger event. Furthermore, we connect the $M_{*}$-$j_{*}$ plane, quantified by the $b$-value, with the $λ_{\mathrm{R}}$-$ε$ plane, revealing a strong correlation between the position of a galaxy in the $λ_{\mathrm{R}}$-$ε$ plane and the $b$-value. Going one step further, we classify our sample based on features in their velocity map, finding all five common kinematic groups, also including the recently observed group of prolate rotators, populating distinct regions in the $λ_{\mathrm{R}}$-$b$ plane.

astro-ph.GA↗

arXiv2018

Ricci flow from spaces with isolated conical singularities

Let $(M,g_0)$ be a compact $n$-dimensional Riemannian manifold with a finite number of singular points, where the metric is asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1},g)$. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. In the case that the initial manifold has isolated singularities asymptotic to a non-negatively curved cone over $(\mathbb{S}^{n-1}/Γ,g)$, where $Γ$ acts freely and properly discontinuously, we extend the above result by showing that starting from such an initial condition there exists a smooth Ricci flow with isolated orbifold singularities.

math.DG↗

arXiv2018

Consequences of strong stability of minimal submanifolds

In this note we show that the recent dynamical stability result for small $C^1$-perturbations of strongly stable minimal submanifolds of C.-J. Tsai and M.-T. Wang directly extends to the enhanced Brakke flows of Ilmanen. We illustrate applications of this result, including a local uniqueness statement for strongly stable minimal submanifolds amongst stationary varifolds, and a mechanism to flow through some singularities of Lagrangian mean curvature flow which are proved to occur by Neves.

math.DG↗

arXiv2018

On short time existence for the planar network flow

We prove the existence of the flow by curvature of regular planar networks starting from an initial network which is non-regular. The proof relies on a monotonicity formula for expanding solutions and a local regularity result for the network flow in the spirit of B. White's local regularity theorem for mean curvature flow. We also show a pseudolocality theorem for mean curvature flow in any codimension, assuming only that the initial submanifold can be locally written as a graph with sufficiently small Lipschitz constant.

math.AP↗