User hapchiu - MathOverflow

Corona Theorem in several variables

2012-11-26T07:16:23Z

Hallo,

I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit disk, which do not all simultaniously vanish. Then there exists bounded holomorphic functions $g_{1}, ..., g_{n}$ such that $\sum_{i=1}^{n}f_{i}g_{i} = 1$. My question is: is this true in several variables? Let say, if the disk is a polydisk or some open, convex ... domain?

hapchiu

$SU(n)$-structures on a manifold

2013-03-19T16:36:41Z

I have the following question. Consider a $2n$-dimensional almost complex manifold $M$. Assume that on $M$ there exists a complex valued $n$-form $\Omega$ and a $2$-form $\omega$ such that:

$\Omega$ is locally decomposable, i.e. there exists $n$ $1$-forms $\theta_{i}$ such that $\Omega = \theta_{1} \wedge ... \wedge \theta_{n}$.
$\Omega \wedge \omega = 0$.
$|\Omega \wedge \overline{\Omega}| > 0$.

How can one show that $M$ admits an $SU(n)$-structure? I think to define charts and show that the transitions functions are elements of $SU(n)$. But how to define the trivialisations ?

Greetings hapchiu

Flat connection, finite-dimensional space of covariant constant one forms

2013-03-03T06:42:04Z

hallo,

I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the space of covariant constant one forms on $U$ is finite dimensional, i.e. $n$-dimensional real vector space ? is it true ? or no ?

hapchiu

Question in the paper of Robert Bryant "Calibrated embeddings in the special Lagrangian and coassociative cases"

2013-02-20T13:27:02Z

Hallo,

I am trying to read the paper "Calibrated embeddings in the special Lagrangian and coassociative cases" by Robert Bryant and I have a question. I hope that maybe one of you could give me some answers. The paper can be found here: http://arxiv.org/abs/math/9912246 . Here is my question: On page 12, in the middle, there are defined the spaces $\mathfrak{h}_{k}$ = { $x\in \mathbb{R}^{n\times n}$ | $\iota^{*}_{k}(x.\alpha)=0$, $\forall \alpha \in $ $\Lambda^{*}(\mathbb{R}^{n})^{G}$ }. I am not very familiar with the notation. I think $x.\alpha$ means the Lie-derivative of $\alpha$ in the direction $x$. Am I right? If so, using the formula for the Lie-derivative $\mathcal{L}_{X}= \iota d + d\iota$. I do not see how to plug in a matrix $x \in \mathbb{R}^{n\times n}$ in a form defined on $\mathbb{R}^{n}$. How can this be understood? I think, by using the $\mathbb{R}^{n}$-valued form $\nu$ defined by $\nu (v) = u(\pi ' (v))$ , for all $v \in T _{u} F$ (also on page 12 in the middle). But still, for $x \in \mathbb{R}^{n\times n} = Lie(GL(n,\mathbb{R}))=ker(\pi ')$ one gets $ \nu(x)=0 $. How is this to be understood? I would be very tankfull if somebody could help me with this.

greetings hapchiu

Uniqueness of Kähler form with same volume

2013-01-11T09:15:15Z

Hallo,

Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $J$ and a Kähler form $-i\partial \overline{\partial} \phi$ such that $M$ is a Lagrangian manifold and $\phi = d\phi =0$ on $M$. This shows the existence. Now lets assume that there are two symplectic forms $\omega_{1} = -i\partial \overline{\partial} \phi_{1}$ and $\omega_{2} = -i\partial \overline{\partial} \phi_{2}$ in the same complex structure and such that $M$ is a Lagrangian manifold with respect to both and $\omega_{1}^{n} = \omega_{2}^{n}$ (they have the same volume), where $n$ is the dimension of $M$. My question is: are these Kähler forms the same? If not, what assumptions does one need to make in order that they are equal? Or, is there any chance, at all, to make them equal?

hapchiu

Solution of a PDE and its uniqueness

2013-01-02T06:49:07Z

Hallo,

consider $f: U \times I \rightarrow \mathbb{R}$, where $U \subset \mathbb{R}^{n}$ and $0 \in I \subset \mathbb{R}$ be two open sets. I am looking for the solution $f$ of the following PDE $\sum_{i=0}^{n} (\frac{\partial^{2}f}{\partial t^{2}})^{i} K_{i}(x,t,f,\frac{\partial f}{\partial t}, \frac{\partial^{2} f}{\partial t \partial x_{j}}, \frac{\partial^{2} f}{\partial x_{k} \partial x_{j}}) = 0$, with initial condition $f(x,0) = \frac{\partial f}{\partial t}(x,0) = 0$. Here the $K_{i}$ are analytic functions depending of several variables and $f$ should also be analytic. The solution should not be on the whole $U \times I$, maybe on some smaller open set. Does there exists a solution? If yes, is this solution unique? When does a solution exists? I have read what bryant wrote and it seems now clearly to me. But I have a different suggestion (I am not 100 % convinced but maybe one can point out where I did a mistake). Here is what I thaught: Differentiate the whole PDE expression with respect to $t$, then one obtains: $\frac{\partial }{\partial t}(\sum_{i=0}^{n} (\frac{\partial^{2}f}{\partial t^{2}})^{i} K_{i}(x,t,f,\frac{\partial f}{\partial t}, \frac{\partial^{2} f}{\partial t \partial x_{j}}, \frac{\partial^{2} f}{\partial x_{k} \partial x_{j}})) = 0$. Thus one obtains $\sum_{i=1}^{n}i(\frac{\partial^{2}f}{\partial t^{2}})^{i-1} \cdot (\frac{\partial^{3}f}{\partial t^{3}})K_{i} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{i}}{\partial t} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{0}}{\partial t} = 0$. Now we write the PDE as: $(\frac{\partial^{3}f}{\partial t^{3}}) \cdot F + G = 0$, where $F = \sum_{i=1}^{n}i(\frac{\partial^{2}f}{\partial t^{2}})^{i-1}K_{i}$ and $G = \sum_{i=1}^{n}(\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{i}}{\partial t} + (\frac{\partial^{2}f}{\partial t^{2}}) \cdot \frac{\partial K_{0}}{\partial t}$. Now if $F$ does not vanish on ${t=0}$ we consider the equation $(\frac{\partial^{3}f}{\partial t^{3}}) \cdot F + G = 0$ and can use Cauchy-Kovalewskaya with appropriate initial conditions. This was my guess. Is this right? Where is the mistake?

hapchiu

Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold

2012-12-07T08:50:44Z

Hallo,

Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler metric on $M$. $g$ is now a Riemannian metric on $A$. Let $U$ be an arbitrary small neigbourhood of $A$ in $M$. Is it possible to embedd $U$ in some $\mathbb{C}^{N}$ isometrically? I think its always possible to embed such an arbitrary small neighbourhood $U$ in $\mathbb{C}^{N}$ for some $N$. But can this be also done isometrically?

hapchiu

Real analytic submanifolds of $\mathbb{R}^{n}$

2012-12-06T07:02:02Z

Hallo,

Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as the zero set of some polynomials? If yes, why ? Are there any references?

Constructing the imaginary part of a holomorphic function

2012-12-04T16:57:55Z

Hallo,

Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function $\varphi : U \rightarrow \mathbb{R}$ such that $f + i \varphi$ is holomorphic? Is this possible?

hapchiu

Kähler form on complex Lie group

2012-12-01T17:48:09Z

Hallo,

Let $G$ be a semi-simple, compact Lie Group. Consider its complexification $G_{\mathbb{C}}$. Does there exist a Kähler structure on $G_{\mathbb{C}}$ which is $G$-invariant (maybe in a neighbourhood of $G$ in $G_{\mathbb{C}}$)?

hapchiu

Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold

2012-11-29T06:35:58Z

Hallo,

It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught of as some open neighbourhood of the zero section of the cotangent bundle $T^{*}M$. This complex manifold is far from being compact. My question is now: Can one choose the complex manifold $\Omega$ not to be just some open neighbourhood of $M$ but to be a compact Kähler manifold? In other words: Can one embed any real-analytic compact Riemannian manifold isometrically in a compact Kähler manifold, such that the real-analytic Riemannian manifold is totally real in the Kähler manifold? Are there any references on this topic? What furter assumptions does one need in order that this works? Or is it entirely impossible? If so, why? By compact I mean compact without boundary!

hapchiu

Analytic Lagrangian Submanifolds

2012-11-17T15:24:22Z

Hallo,

I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there similar ones? I am actually interested in understanding better the construction of a defining phase function for a Lagrangian submanifold and to understand the uniqueness. Is there any other literature on that? Actually I am interested in the proof of the following: Let $M$ be a connected Lagrangian submanifold of a Kähler manifold $\Omega$, then there is a neighbourhood $U$ of $M$ in $\Omega$ and a unique defining phase function $\phi$ on $U$ for $M$.

hapchiu

Polarisation in a nighbourhood of a Lagrangian submanifold

2012-11-13T13:31:44Z

Hallo,

Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a Lagrangian submanifold such that $\alpha = 0$ of $TX|_{M}$. I am interested in the following questions:

Is there a unique polarisation defined on $X$ near $M$ which is transversal to $M$ and whose one form is $\alpha$ ? By polarisation I mean the following: A polarisation of a symplectic manifold $X$, with symplectic form $\omega$, is a smooth assignment of a Lagrangian subspace of $T_{x}X$ to each $x \in X$ in such a way that this assignment is integrable.
If 1. is true, is there a symplectic diffeomorpism $\Phi$ of a neigbourhood of $M$ is $X$ with a neigbourhood of $M$ in its cotangent bundle which carries the leaves of the polarisation into the standard cotangent fibration of $T^{*}M$ ?

Actually I know that these results are true. I would like to see the proof of them. Are there any references where I can look them up? If so, can you please tell me where these references can be found? Thanks a lot!

hapchiu

Harmonic Function?

2012-11-11T15:05:28Z

Hi,

Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + iy_{1}, ..., x_{n} + iy_{n})} \sqrt{\varphi (\xi)} d \xi)$ harmonic. Where integration should be understood: integrate along any path from $0$ to $(x_{1} + iy_{1}, ..., x_{n} + iy_{n})$ (by Cauchy the integral is path independet). With harmonic I mean: $\Delta u = \frac{\partial^{2}u}{\partial x_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial x_{n}^{2}} + \frac{\partial^{2}u}{\partial y_{1}^{2}} + ... + \frac{\partial^{2}u}{\partial y_{1}^{n}} = 0$. I was reading about this and in 2 dimensions it is harmonic. Now I am actually interested if its also harmonic in more dimensions. Is it?

hapchiu

Rotation in Hyperkähler manifolds

2012-11-06T05:10:23Z

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) \in S^{2} \subset \mathbb{R}^{3}$? Hope the question is not too trivial :).

Answer by hapchiu for example of special lagrangian submanifold

2012-03-12T09:49:10Z

I think, eaven in this case you mentioned there counterexamples, but cannot be given explicitely. See the post of Bryant (above), part 1. Am I right?

hapchiu

Comment by hapchiu

2013-03-03T07:51:33Z

anyone has an idea ?

Comment by hapchiu

2013-02-20T14:00:31Z

is there any interpretation of this using the Lie-derivative ?

Comment by hapchiu

2013-02-20T13:43:04Z

i have edited my question once again so one actually can read it :).

Comment by hapchiu

2013-01-12T09:14:14Z

@Bryant: I take any complex structure, since by Bruhat-Whitney in a neighbourhood of the zero section they are biholomorphic.

Comment by hapchiu

2013-01-02T12:38:04Z

what do you mean by characteristic?

Comment by hapchiu

2013-01-02T09:13:11Z

why do you think so?

Comment by hapchiu

2012-12-07T13:29:07Z

I thaught that one can do the following: since $A$ is real analytic we can use the real analytic version of Nash embedding theorem and consider $A$ as a real analytic Riemannian submanifold of some $\mathbb{R}^{N}$. Then locally $A$ is the zero set of some real analytic functions. Extend these functions holomorphically and then patch them together, since on the overlaps of some open sets in $A$ these real analytic functions are the same, and the extension of them would be the same holomorphic function. Is this possible?

Comment by hapchiu

2012-12-07T13:18:53Z

yes, is it possible if the embedding is holomorphic?

Comment by hapchiu

2012-12-04T17:28:00Z

in dimension $\geq 2$? does the same still hold?

Comment by hapchiu

2012-12-02T08:59:56Z

what do you mean by average its pullbacks under left and right multiplications if $G$ is compact???

Comment by hapchiu

2012-12-02T08:27:31Z

Is there any reference where I can see how the proof is done?

Comment by hapchiu

2012-11-29T09:27:42Z

If all this works is the resulting complex manifold then compact (compact without boundary)?

Comment by hapchiu

2012-11-29T09:12:49Z

Thanks for the answer Dmitri. Is there any reference on this?

Comment by hapchiu

2012-11-27T11:33:23Z

why does this paper not solve the posted problem?

Comment by hapchiu

2012-11-27T11:26:15Z

what do you mean by "BMO"?