Related papers: On smooth maps with finitely many critical points

In this paper, we study diagonal maps between spheres given by two homogeneous polynomial maps between spheres, defined on the same domain sphere. First we find their bitension field, then we give a method for generating proper biharmonic…

The purpose of this paper is to present simple and fast methods for computing control points for polynomial curves and polynomial surfaces given explicitly in terms of polynomials (written as sums of monomials). We give recurrence formulae…

We provide an algorithm of constructing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space. Our algorithm returns a reasonably short curve between two…

We describe how to compute topological objects associated to a polynomial map of several complex variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the…

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if…

We construct equivariant harmonic maps between cohomogeneity one manifolds.

We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of…

Let f be a polynomial or a rational function which has r summable critical points. We prove that there exists an r-dimensional manifold in an appropriate space containing f such that for every smooth curve in it through f, the ratio between…

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…

In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes…

In this note we prove the existence of two proper biharmonic maps between the Euclidean ball of dimension bigger than four and Euclidean spheres of appropriate dimensions. We will also show that, in low dimensions, both maps are unstable…

Both bi-harmonic map and $f$-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study $f$-bi-harmonic maps as the critical points of the $f$-bi-energy…

The onset and bifurcation points of the $n$-cycles of a polynomial map are located through a characteristic equation connecting cyclic polynomials formed by periodic orbit points. The minimal polynomials of the critical parameters of the…

Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function…

We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from…

We compute the $k$-width of a round $2$-sphere for $k=1,\ldots,8$ and we use this result to show that unstable embedded closed geodesics can arise with multiplicity as a min-max critical varifold.

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the…

In this article, we investigate the geometry of critical metrics of the volume functional on an $n$-dimensional compact manifold with (possibly disconnected) boundary. We establish sharp estimates to the mean curvature and area of the…

Random plane wave is conjectured to be a universal model for high-energy eigenfunctions of the Laplace operator on generic compact Riemanian manifolds. This is known to be true on average. In the present paper we discuss one of important…