mg.metric geometry – Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim?

Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from that vertex.Then, center of excircle of this triangle which touches side of pentagon, the vertex of pentagon opposite to that side and the incenter of the pentagon formed by diagonals are collinear.

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GeoGebra applet that demonstrates this claim can be found here.

ag.algebraic geometry – Extending rational maps to semi-abelian varieties

Setup. Let $k$ be an algebraically closed field of characteristic zero, and let $G/k$ be a semi-abelian variety i.e., $G$ is a commutative algebraic group which is an extension of an abelian variety $A/k$ by a torus $T/k$, so it admits the following presentation:
$$0 to T to G to A to 0$$
Let $Z/k$ be a smooth, integral variety and let $Usubset Z$ be a dense open such that $text{codim}(Zsetminus U) geq 2$.

Question. Is it true that any morphism of $k$-schemes $Uto G$ uniquely extends to a morphism $Zto G$?

This result is stated in Lemma A.2 of Mochizuki’s Topics in absolute anabelian geometry I: generalities and the proof goes as follows. First, he deduces the result when $G$ is proper i.e., $G$ is isomorphic to $A$. In this case the result is well-known and can be proved in a few different ways; one avenue of proof uses that $A$ does not contain any rational curves. Next, he says that we may reduce to the case where $G$ is isomorphic to a torus. Again, this result is well-known and follows from an application of Serre’s normality criterion.

I understand the proofs of both cases, but I am struggling to see how to combine these to get the result for a general semi-abelian variety $G/k$. More precisely, I do not understand how one may reduce to the case where $G$ is a torus.

Any comments, suggestions, references, and/or counter-examples would greatly be appreciated!

ag.algebraic geometry – Which infinite-dimensional Lie algebras have realizations as algebras of global sections of vector bundles with special structure?

I would not be surprised by downvotes since this question is at the same time very naïve, very vague and might be asking about things well known for decades. Specifically vagueness comes from the word “special” in the title. As for that third weakness, I have a weak defence against it by choosing the reference request tag.

The starting point is the trivial observation that for a finite-dimensional semisimple Lie algebra $mathfrak g$, the algebra $mathfrak g(t,t^{-1})$ of Laurent polynomials with coefficients in $mathfrak g$ can be viewed as the algebra of polynomial global sections of the trivial vector bundle with fibre $mathfrak g$ over the punctured plane $mathbb C^times$.

My first question is whether it is possible to somehow modify this bundle in such a way that the global sections will turn out to be the affine Lie algebra $hat{mathfrak g}$ (the universal central extension of $mathfrak g(t,t^{-1})$); similarly for twisted versions of $hat{mathfrak g}$.

Second question – is it known what does one obtain with nontrivial vector bundles, and with some projective curve in place of $mathbb C^times$? I have no idea whether one indeed obtains a Lie algebra at all or something different. Does the algebraic group structure of $mathbb C^times$ have any significance in this context? Accordingly, is it important whether I take an elliptic curve or curves of all genera give more or less similar results?

Finally, I remember that although sections of the tangent bundle form a Lie algebra, it is not a Lie algebra bundle in the sense that fibres do not have any Lie algebra structure. Still, if I am not mistaken, the Virasoro algebra is the universal central extension of the algebra of global sections of a tangent bundle, right? Again, is there some modified form of the tangent bundle such that the Virasoro algebra would be global sections of this modified bundle? And again, is the group structure significant here? What are algebras of sections of tangent bundles over projective curves, in particular, over elliptic curves? Do they have nontrivial central extensions?

geometry – Minimizing Integral Of Distances To The Power Of $p$

I read about different centers for measurable compact sets $S$ in $mathbf{R}^n$ and wondered if there is more. As far as I understood, the center of mass can be defined as the point $x^*$ minimizing $int_{xin S}|x-x^*|^2$, but what about $int_{xin S}|x-x^*|^p$, does this also have a unique minimum?

I could prove for finite sets (taking sum instead of integral) that there is a unique minimum if not all points lie on a line and $pge1$, I assume this generalizes to the integral as well. With $p<1$ it makes sense that there is no unique minimum in the finite case, i.e. take three equidistant points and very small p, then clearly the minimum is attained at each of those three points but not in between.

But I wonder if the minimum is maybe unique again in the integral case only for $0<p<1$ as well?

dg.differential geometry – largest geodesic ball inside a small portion of Euclidean submanifold

Suppose that $Msubseteqmathbb R^D$ is a compact smooth Riemannian submanifold of dimension $d$, having normal injectivity radius $tau$. Let $x_0in M$ be a point, and $deltain (0,tau)$ sufficiently small. I would like to know if $Mcap B_{mathbb R^D}(x_0,delta)$ has $d$-dimensional Hausdorff measure at least $(1/2)delta^domega_d$ where $omega_d$ is the volume of unit ball in $mathbb R^d$.

My approach: since $deltain (0,tau)$, there is a diffeomorphism $exp_{x_0}:B_{T_{x_0}M}(0,delta)cong B_M(x_0,delta)$. Now, by Taylor expansion of the volume form (theorem 9.9/9.10 of Gray’s Tubes) we know, for example, that volume of $B_M(x_0,delta)$ is more than $0.9$ times the volume of $B_{mathbb R^d}(0,delta)$; hence, if I know that $B_M(x_0,delta)subseteq Mcap B_{mathbb R^D}(x_0,delta)$, for example, then I will be done.

differential geometry – A polygon with constant angular momentum bounds a circle

Let $alpha:(0,L) to mathbb{R}^2$ be a piecewise affine map satisfying $alpha(0)= alpha(L)$ and $|dot alpha|=1$. Supopse that $alpha(t) times dot alpha(t)$ is constant.

How to prove that $operatorname{Image}(alpha)$ is a tangential polygon, i.e. a polygon whose edges are all tangent to a fixed circle, centered at the origin?

It suffices to prove that for each subinterval $(a,b) subseteq (0,L)$ where $alpha|_{(a,b)}$ is affine, there exists a $t_0 in (a,b)$ such that $dot alpha(t_0) perp alpha(t_0)$.

Indeed, if this is the case, then $|alpha(t_0)|=|alpha(t) times dot alpha(t)|=C$ is independent of the segment $(a,b)$ chosen. Thus, every “edge” $alpha((a,b))$, contains a point $P_{a,b}=alpha(t_0)$ on the circle with radius $C$, and the edge is perpendicular to radius at $P_{a,b}$, i.e. it is tangent to the circle at $P_{a,b}$.

I am not sure how to prove the bold statement. I think we need to use somehow the fact that the polygon “closes”.

The converse implication is easy:

If there exists such a circle with radius $R$, then $|alpha(t) times dot alpha(t)|=R$ is constant: Indeed, suppose that $alpha(t_0)$ lies on the circle — so it is a tangency point.

Then $dot alpha(t_0) perp alpha(t_0)$, and $|alpha(t_0)|=R$.

Let $t$ satisfies $dot alpha(t)=dot alpha(t_0)$, i.e. $alpha(t)$ belongs to the same edge as $alpha(t_0)$. Then

$alpha(t)=alpha(t_0)+beta(t)$, where $beta(t) || dot alpha(t_0)$, so
alpha(t) times dot alpha(t)=big( alpha(t_0)+beta(t) big) times dot alpha(t_0)=alpha(t_0) times dot alpha(t_0),

which implies $|alpha(t) times dot alpha(t)|=R$.

ag.algebraic geometry – Vanishing of intermediate cohomology for a multiple of a divisor

Let $S subset mathbb P^3$ be a smooth projective surface (over complex numbers). Let $C$ be a smooth hyperplane section. Let $Delta$ be a non-zero effective divisor on $S$ such that $h^1(mathcal O_S(nC+Delta))=0, h^1(mathcal O_S(nC-Delta))=0$ for all $n in mathbb Z$. Then my question is the following :

In this situation can we say that: $h^1(mathcal O_S(m Delta))=0$ for $m geq 2$? Can we impose any condition so that this happens?

Any help from anyone is welcome.

ag.algebraic geometry – Algebraic properties of geodesics

This is a question related to my last post. I will use the same definition here.

A complete smooth manifold $M$ with an affine connection $nabla$ is said to have an algebraic model of dimension $n$ if there exists a smooth immersion $sigma:M rightarrow Bbb{R}^n$ such that each image of geodesics on $M$ with respect to $nabla$ is either sub-algebraic or improper. A sub-algebraic set is a subset of $Bbb{R}^n$ defined by the common zeros and positive or non-negative parts of finitely many polynomials. (For example, half circles and toroidal handles are sub-algebraic sets.) In this sense, all simply connected hypaerbolic spaces have algebraic models (e.g. upper half space, Poincare’s unit $n$-ball). Note that I do not require $sigma$ to be algebraic — the usual hyperbolic cases are apparently not algebraic.

The first question arises naturally on how ‘algebraic’ $sigma(M)$ really is. I conjectured that if $M$ has an algebraic model $sigma$ then $sigma(M)$ is sub-algebraic — more specifically, the ‘infinite boundary’ of $sigma(M)$ is an algebraic set. It seems a counterexample is not easy to construct.

The second question concerns about a certain type of manifolds, namely the 3-manifolds with geometric structures in Thurston’s sense. I want to know if there is an algebraic model of at least 1 manifold of each of the 8 types. And as they have Lie groups as transition groups, I also conjectured that every geometric 3-manifold has an algebraic model. (The corresponding 8 Lie groups all seem to have algebraic models.) I am looking forward to any reference articles on this problem.

geometry – What Height can Divide the Area of a Triangle by 2

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dg.differential geometry – Uniform convergence of Eigenfunction decomposition on Riemannian sphere?

Let ${u_k}_{k=1}^infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-Delta$ on the sphere $mathbb{S}^n$ (here $Delta$ is the Laplace Beltrami operator). Let $u$ be a smooth (real valued) function on the sphere. It is a well-known result that we can write $u=sum_{k=1}^infty c_k u_k$ for some (real) constants $c_k$. My question is: Is the convergence of this sum uniform?

I am trying to prove that the optimal constant in the Poincare inequality is $lambda_1=n$. That is to say, I am trying to prove the inequlity $int_{mathbb{S}^n} |nabla u|^2 geq n int_{mathbb{S}^n} |u|^2$. Here is what I have done so far:

First, integrate by parts on the LHS so that it suffices to prove $int_{mathbb{S}^n} -uDelta u geq n int_{mathbb{S}^n} |u|^2$. Then use $u=sum_{k=1}^infty c_k u_k$ and assume that the convergence is uniform. Then we can switch the order of the sum with the derivative and integral (and use the fact that ${u_k}$ are orthonomal) so that
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)Delta left(sum_{j=1}^infty c_j u_jright)&=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty c_j Delta u_jright)=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty lambda_j c_j u_jright)
\&= sum_{j,k}c_k c_j lambda_jint_{mathbb{S}^n} u_k u_j= sum_{j,k}c_k c_j lambda_j delta_{jk}=sum_{j}c_j^2 lambda_jgeq lambda_1 sum_j c_j^2.

By the same logic, the last sum is equal to $int |u|^2$.

Now obviously, this proof requires some argument showing that the sum commutes with $Delta$ and the integral but I have not been able to find a reference that the sum converges uniformly. My thought is that this would follow from some basic facts in Harmonic analysis though I am no expert in that field. Would anyone be able to provide a reference for this?