arXiv research

Unstable minimal surfaces are the unstable stationary points of the Dirichlet-Integral. In order to obtain unstable solutions, the method of the gradient flow together with the minimax-principle is generally used. The application of this method for minimal surfaces in the Euclidean spacce was presented in \cite{s3}. We…

Noncompact Ricci-flat solutions have infinite unstable dimensions.

We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.

In this paper, we introduce a non linear ODE method to construct CMC surfaces in Riemannian manifolds with symmetry. As an application we construct unstable CMC spheres and outlying CMC spheres in asymptotically Schwarzschild manifolds with metrics like $g_{ij}=(1+\frac{1}{l})^{2}δ_{ij}+O(l^{-2})$. The existence of uns…

We show that the pair $(X, -K_X)$ is K-unstable for a del Pezzo manifold $X$ of degree five with dimension four or five. This disprove a conjecture of Odaka and Okada.

We show that a strict, nearly Kähler $6$-manifold with either second or third Betti number nonzero is linearly unstable with respect to the $ν$-entropy of Perelman and hence is dynamically unstable for the Ricci flow.

The paper proves regularity of states on manifolds with unstable dynamics.

In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain …

Study finds Kähler-Einstein metrics on two Pasquier varieties.

Proves transitivity of real Anosov diffeomorphisms with specific properties.

New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.

We construct Hodge filtered function spaces associated to infinite loop spaces. For Brown-Peterson cohomology, we show that the corresponding Hodge filtered spaces satisfy an analog of Wilson's unstable splitting. As a consequence, we obtain an analog of Quillen's theorem for Hodge filtered Brown-Peterson cohomology fo…

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based…

In this article, we thoroughly investigate the stability inequality for Ricci-flat cones. Perhaps most importantly, we prove that the Ricci-flat cone over CP^2 is stable, showing that the first stable non-flat Ricci-flat cone occurs in the smallest possible dimension. On the other hand, we prove that many other example…

Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with $π_1(M) = Z^4times Z/3$, so that the index invariant in the KO-theory of the reduced $C^*$-algebra of $π_1(M)$ is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of pos…

Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DW…

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…

The paper explores maximal destabilizers for both K-stability and Chow-stability in unstable situations.

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by $\mathcal{R}_{\frac{n}{2}}(g):= \int_M |R(g)|^{\frac{n}{2}}dv_g$ where $R(g)$, $dv_g$ denote the Riemannian curvature and volume form corresponding to $g$. We show that there are lo…

We refine a metric bunching estimate for pinched manifolds.

The linear stability of warped product Einstein metrics as fixed points of the Ricci flow is investigated. We generalise the results of Gibbons, Hartnoll and Pope and show that in sufficiently low dimensions, all warped product Einstein metrics are unstable. By exploiting the relationship between warped product Einstei…

Study explores unstable 3-forms on Calabi-Yau 3-folds.

Nearly $G_2$-structures are unstable under a modified $G_2$-Laplacian co-flow.

Ricci flow simulations show unstable Fubini-Study metrics develop singularities.

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

Paper shows how to transform certain flows into R-covered ones.

We study time-like hypersurfaces with vanishing mean curvature in the (3+1) dimensional Minkowski space, which are the hyperbolic counterparts to minimal embeddings of Riemannian manifolds. The catenoid is a stationary solution of the associated Cauchy problem. This solution is linearly unstable, and we show that this …

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao …

We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

In this paper, we motivate and define $Φ$-energy density, $Φ$-energy, $Φ$-harmonic maps and stable $Φ$-harmonic maps. Whereas harmonic maps or $p$-harmonic maps can be viewed as critical points of the integral of $σ_1$ of a pull-back tensor, $Φ$-harmonic maps can be viewed as critical points of the integral of $σ_2$ of…

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps $f \colon U \subseteq \mathbb{R}^d \to \mathbb{R}^d$ by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index…

Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.

New loss function helps learn unstable dynamical systems.

Unstable minimal surfaces in n-space link to hyperbolic products.

An asymptotic formula for the Tian-Paul CM-line of a flat family blown-up at a flat closed sub-scheme is given. As an application we prove that the blow-up of a polarized manifold along a (relatively) Chow-unstable submanifold admits no (extremal) constant scalar curvature Kahler metrics in classes making the exception…

Segre varieties' hyperplane sections are unstable under certain conditions.

Given for instance a finite volume negatively curved Riemannian manifold $M$, we give a precise relation between the logarithmic growth rates of the excursions into cusps neighborhoods of the strong unstable leaves of negatively recurrent unit vectors of $M$ and their linear divergence rates under the geodesic flow. As…

We show that a 1-parameter family Ricci flow ancient solutions arises from the natural collapsings of the twistor space of positive quaternion Kähler manifolds. We use these ancient solutions to show that a positive quaternion Kähler manifold is isometric to one of the Wolf spaces.

Study of flows on complex manifolds with holomorphic properties.

In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (…

Study shows instability of specific cone solutions in high-dimensional spaces.

This paper is a short version of some joint work with Stefan Haller. It describes the structure of "smooth manifold with corners" on the space of possibly broken instantons and on the completion of unstable manifolds of a generic smooth vector field. The result is stated in Theorem 1.4.

Solves complex Hessian equations in unstable cases, proving unique canonical solutions with singularities.

In this note we show that the bi-invariant Einstein metric on the compact Lie group $G_{2}$ is dynamically unstable as a fixed point of the Ricci flow. This completes the stability analysis for the bi-invariant metrics on the compact, connected, simple Lie groups. Interestingly, $G_{2}$ is the only unstable exceptional…

Algorithm identifies and transfers unstable features to create robust classifiers.

Survey of computations for Riemann surface moduli spaces, focusing on unstable homology.