Related papers: $L^2$ affine Fourier restriction theorems for smoo…
We consider a surface with negative curvature in $\Bbb R^3$ which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific…
In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of…
We study the descent behaviour of homotopy-theoretic properties of smooth complex affine surfaces under finite surjective morphisms. We first examine the Eilenberg-MacLane property and show, by means of an explicit counterexample, that it…
In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to…
A method is presented for computing all the affine equivalences between two rational ruled surfaces defined by rational parametrizations that works directly in parametric rational form, i.e. without computing or making use of the implicit…
Let $\phi$ be a smooth function on a compact interval $I$. Let $$\gamma(t)=\left (t,t^2,\cdots,t^{n-1},\phi(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(\gamma(t))\big|^q \big|\phi^{(n)}(t)\big|^{\frac{2}{n(n+1)}}…
In this paper we prove a uniform Fourier restriction estimate over the class of simple curves where the last coordinate function can be extended to a holomorphic function of bounded frequency in a sufficiently large disc. The proof is based…
We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma, where the real condition of sufficient smoothness for the phase is replaced by the condition that the phase is a convergent power series. This van der Corput style…
New sharp affine isoperimetric inequalities for volume decomposition functionals $X_{2}$ and $X_{3}$ in $\mathbb{R}^n$ are established. To fulfil this task, we prove the recursion formulas for volume decomposition functionals and find out…
This paper explores the Bernstein problem of smooth maps $f:\mathbb{R}^4 \to \mathbb{R}^3$ whose graphs form coassociative submanifolds in $\mathbb{R}^7$. We establish a condition, expressed in terms of the second elementary symmetric…
In this paper, we study the restriction estimate for a certain surface of finite type in $\mathbb{R}^3$, and partially improves the results of Buschenhenke-M\"{u}ller-Vargas. The key ingredients of the proof include the so called…
A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3,…
A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…
In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…
A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and…
We classify smooth surfaces whose higher cohomologies of i-forms for all i vanish. We show that if such a surface is not affine, then it has essentially two possibilities.
We consider the class $PW(\mathbb R^n)$ of functions in $L^2(\mathbb R^n)$, whose Fourier transform has bounded support. We obtain a description of continuous maps $\varphi : \mathbb R^m\rightarrow\mathbb R^n$ such that $f\circ\varphi\in…
This paper is devoted to the study of affine quaternionic manifolds and to a possible classification of all compact affine quaternionic curves and surfaces. It is established that on an affine quaternionic manifold there is one and only one…
Given a closed, oriented surface, possibly with boundary, and a mapping class, we obtain sharp lower bounds on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, both with and…
Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f in R, we bound the degree of a nonzero homogeneous sum-of-squares g in R such that the product fg is again a sum of squares. Better…