Geometric and Functional Analysis
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Stable Norms of Surfaces: Local Structure of the Unit Ball at Rational Directions
Geometric and Functional Analysis - Tập 7 - Trang 996-1010 - 1997
Let (M, g) be a closed orientable surface, equipped with a smooth Finsler
metric. The metric induces a norm on the real homology of M, called a stable
norm. We show this norm is neither strictly convex, nor smooth.
Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group
Geometric and Functional Analysis - Tập 8 Số 5 - Trang 853-931 - 1998
The Cut Loci, Conjugate Loci and Poles in a Complete Riemannian Manifold
Geometric and Functional Analysis - Tập 22 - Trang 1400-1406 - 2012
Let M be a complete Riemannian manifold. We first prove that there exist at
least two geodesics connecting p and every point in M if the tangent cut locus
of $${p \in M}$$ is not empty and does not meet its tangent conjugate locus. It
follows from this that if M admits a pole and $${p \in M}$$ is not a pole, then
the tangent conjugate and tangent cut loci of p have a point in common. Here we
say t...
Flattening a non-degenerate CR singular point of real codimension two
Geometric and Functional Analysis - Tập 28 - Trang 289-333 - 2018
This paper continues the previous studies in two papers of Huang–Yin [HY16,HY17]
on the flattening problem of a CR singular point of real codimension two sitting
in a submanifold in $${{\mathbb {C}}^{n+1}}$$ with n + 1 ≥ 3, whose CR points
are non-minimal. Partially based on the geometric approach initiated in [HY16]
and a formal theory approach used in [HY17], we are able to provide more or less
...
An isoperimetric inequality for the Heisenberg groups
Geometric and Functional Analysis - Tập 8 - Trang 219-233 - 1998
We show that the Heisenberg groups $ \cal {H}^{2n+1} $ of dimension five and
higher, considered as Riemannian manifolds, satisfy a quadratic isoperimetric
inequality. (This means that each loop of length L bounds a disk of area ~ L 2.)
This implies several important results about isoperimetric inequalities for
discrete groups that act either on $ \cal {H}^{2n+1} $ or on complex hyperbolic
space, a...
Polynomial bound for partition rank in terms of analytic rank
Geometric and Functional Analysis - Tập 29 - Trang 1503-1530 - 2019
Let $$G_1, \ldots , G_k$$ be vector spaces over a finite field $${\mathbb {F}} =
{\mathbb {F}}_q$$ with a non-trivial additive character $$\chi $$ . The analytic
rank of a multilinear form $$\alpha :G_1 \times \cdots \times G_k \rightarrow
{\mathbb {F}}$$ is defined as $${\text {arank}}(\alpha ) = -\log _q \mathop
{\mathbb {E}} _{x_1 \in G_1, \ldots , x_k\in G_k} \chi (\alpha (x_1,\ldots ,
x_k))$$...
Constant mean curvature hypersurfaces condensing on a submanifold
Geometric and Functional Analysis - Tập 16 - Trang 924-958 - 2006
On the random Chowla conjecture
Geometric and Functional Analysis - Tập 33 - Trang 749-777 - 2023
We show that for a Steinhaus random multiplicative function $$f:{\mathbb
{N}}\rightarrow {\mathbb {D}}$$ and any polynomial $$P(x)\in {\mathbb {Z}}[x]$$
of $$\deg P\ge 2$$ which is not of the form $$w(x+c)^{d}$$ for some $$w\in
{\mathbb {Z}}$$ , $$c\in {\mathbb {Q}}$$ , we have $$\begin{aligned}
\frac{1}{\sqrt{N}}\sum _{n\le N} f(P(n)) \xrightarrow {d} {{\mathcal
{C}}}{{\mathcal {N}}}(0,1), \end{a...
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
Geometric and Functional Analysis - Tập 24 - Trang 946-958 - 2014
We prove that the set of exceptional $${\lambda\in (1/2,1)}$$ such that the
associated Bernoulli convolution is singular has zero Hausdorff dimension, and
likewise for biased Bernoulli convolutions, with the exceptional set independent
of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres
and Schlag, and Hochman. A theorem of this kind is also obtained for
convolutions of h...
Coniveau 2 Complete Intersections and Effective Cones
Geometric and Functional Analysis - Tập 19 - Trang 1494-1513 - 2009
Griffiths computation of the Hodge filtration on the cohomology of a smooth
hypersurface X of degree d in $${\mathbb{P}^n}$$ shows that it has coniveau ≥ c
once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology
of X is then supported on a closed algebraic subset of codimension at least c.
This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a
geometr...
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