Non-hyperbolic surfaces having all ideal triangles with finite area

We construct examples ofC ³ compact surfaces of non-positive curvature having non-Anosov geodesic flows and satisfying the following property: there existsL>0 such that the area of every ideal triangle in the universal covering of the surface is bounded above byL.Partially supported by CNPq of Brazilian Government     

Gaussian mean curvature flow

In the Euclidean Space \mathbb Rⁿ⁺¹{\mathbb {R}^{n+1}} with a density e^{e\frac12 n m2 |x|2}, (e = ±1){e^{\varepsilon \frac12 n \mu^2 |x|^2},} {(\varepsilon =\pm1}), we consider the flow of a hypersurface driven by its mean curvature associated to this density. We give a detailed account of the evolution of a convex hypersurface under this flow. In particular, when e = -1{ \varepsilon=-1} (Gaussian density), the hypersurface can expand to infinity or contract to a convex hypersurface (not necessarily a sphere) depending on the relation between the bound of its principal curvatures and μ.     

Mappings preserving area 1 of triangles

This paper is concerned with the characterization of area-preserving mappings of real inner product spaces.     

Gaussian curvature on singular surfaces

We consider prescribing Gaussian curvature on surfaces with conical singularities in both critical and supercritical cases. First we prove a variant of Kazdan-Warner type necessary conditions. Then we obtain sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai singular metric. We only require that the values of the function are not too large at singular points of the metric with the smallest angle, say, less or equal to 0, or less than its average value. To prove the results, we apply some new ideas and techniques. One of them is to estimate the total curvature along a certain minimizing sequence by using the “Distribution of Mass Principle” and the behavior of the critical points at infinity.     

The Hilbert area of inscribed triangles and quadrilaterals

Geometriae Dedicata - In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary,...     

Pairs of theonian triangles with common area

We study the properties of pairs of triangles with integer sides whose common area is the square of a natural number.     

Gaussian curvature in the negative case

In this paper, we reinvestigate an old problem of prescribing Gaussian curvature in the negative case.

In 1974, Kazdan and Warner considered the equation

on any compact two dimensional manifold with . They showed that there exists a number , such that the equation is solvable for every \alpha > \alpha_o$"> and it is not solvable for .

Then one may naturally ask:

Is the equation solvable for ?

In this paper, we answer the question affirmatively. We show that there exists at least one solution for .

     

On Peterson surfaces of constant Gaussian curvature

The surfaces of constant Gaussian curvature bearing conjugate networks of conic lines are found.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 3–5.     

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta ₁,b ₁,c ₁,A ₁ anda ₂,b ₂,c ₂,A ₂ be the sides and areas of two triangles. Ifa=(a ₁ ^p +a ₂ ^p )^1/p ,b=(b ₁ ^p +b ₂ ^p )^1/p ,c=(c ₁ ^p +c ₂ ^p )^1/p , and 1p4, thena, b, c are the sides of a triangle and its area satisfiesA ^p/2A ₁ ^p/2 +A ₂ ^p/2 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA _f, which satisfies (4A _f/3)^1/2f((4A/3)^1/2).     

On square integrability of mean curvature for surfaces with positive Gaussian curvature

We show that {ie319-1} H ²dµ = for any complete surface M R ³ which has positive curvature outside a compact subset of R ³. This proves a conjecture of Friedrich.     

A characterization of spaces of constant curvature by minimum covering radius of triangles

In this paper, we prove that if in a Riemannian manifold, the minimum covering radius of a point triple of small diameter depends only on the geodesic distances between the points, then the manifold must be of constant curvature. This implies that if in a complete connected Riemannian manifold, the volume of the intersection of three small geodesic balls of equal radii depends only on the distances between the centers and the radius, then it is one of the simply connected spaces of constant curvature. This generalizes an earlier result of the first author and D. Kunszenti-Kovács (2010).     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Decouplings for curves and hypersurfaces with nonzero Gaussian curvature

We prove two types of results. First we develop the decoupling theory for hypersurfaces with nonzero Gaussian curvature, which extends our earlier work from [4]. As a consequence of this, we obtain sharp (up to ε losses) Strichartz estimates for the hyperbolic Schrödinger equation on the torus. Our second main result is an l ² decoupling for nondegenerate curves, which has implications for Vinogradov’s mean value theorem.     

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

A problem of prescribing Gaussian curvature on

A class of functions and the corresponding solutions of

are obtained as a special case of the solutions of

where is defined as .