Related papers: Monotonic invariants under blowups
A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on (C3, 0), without saddle-nodes, has invariant surface. We extend the argument of Cano-Cerveau, done for the…
Conformal invariance of two-dimensional variational problems is a condition known to enable a blow-up analysis of solutions and to deduce the removability of singularities. In this paper, we identify another condition that is not only…
We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is non-trivial due to folding of the image…
Let $G=(V,E)$ be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and uniform polynomial volume growth of degree $m$, all non-negative solutions of the equation…
We give an algebraic way of distinguishing the components of the exceptional strata of quadratic differentials in genus three and four. The complete list of these strata is (9, -1), (6,3,-1), (3,3,3, -1) in genus three and (12), (9,3),…
We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency…
We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the…
We construct a monotone quantity for the classical obstacle problem with non-smooth obstacle, and show that the blow-ups are homogeneous functions of degree $\alpha<2$.
We first present the mixed Hilbert-Samuel multiplicities of analytic local rings over \mathbb{C} as generalized Lelong numbers and further represent them as intersection numbers in the context of modifications. As applications, we give…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations are considered. It is shown that if the inflow is rapidly increasing (pushy) in time, the corresponding laminar profile of the incompressible Euler flow is…
We prove a lower bound for the Milnor number of function germ invariant with respect to a finite abelian group action. It is shown that this bound is tight for functions of arbitrarily many variables. We also prove the function germs that…
We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais-Smale sequences we determine precise blow up rates for subcritical solutions: in…
In the present treatise, a stability analysis of the bottom boundary layer under solitary waves based on energy bounds and nonmodal theory is performed. The instability mechanism of this flow consists of a competition between streamwise…
As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…
It is the aim of this article to determine curvature quantities of an arbitrary Riemannian monotone metric on the space of positive matrices resp. nonsingular density matrices. Special interest is focused on the scalar curvature due to its…
Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new, and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically…
In this paper new numerical invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle…
Shortly after the introduction of Seiberg-Witten theory, LeBrun showed that the sign of the Yamabe invariant of a compact K\"ahler surface is determined by its Kodaira dimension. In this paper, we show that LeBrun's Theorem is no longer…
We investigate in this article the long-time behaviour of the solutions to the energy-dependant, spatially-homogeneous, inelastic Boltzmann equation for hard spheres. This model describes a diluted gas composed of hard spheres under…
For a projective nonsingular curve of genus $g$, the Brill-Noether locus $W^r_d(C)$ parametrizes line bundles of degree $d$ over $C$ with at least $r+1$ sections. When the curve is generic and the Brill-Noether number $\rho(g,r,d)$ equals…