Questions tagged [pseudo-holomorphic-curves]
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48 questions
2
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Is a simple J-holomorphic curve injective everywhere except for finitely many points?
Let $(M^{2n},J)$ be an almost complex manifold and $(\Sigma,j)$ a closed Riemann surface. Suppose $u: \Sigma \to M$ is a simple, nonconstant, $J$-holomorphic curve. Can we prove that the set
$$
Z:=\{z\...
- 1,441
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0
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82
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Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
- 285
4
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92
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A couple of questions about the moduli space of annuli with some marked points on the boundary components
I'm trying to work out an answer for my previous question and I'm stuck with the following issue:
In the paper Deformations of Bordered Riemann surfaces and associahedral polytopes by Devadoss, Heath ...
- 2,018
3
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0
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318
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Bubbling off a sphere in a splitting/stretching manifold
This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology ...
- 2,018
2
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1
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114
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Question on Gromov-Witten invariants when $A=0$
I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's ...
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0
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211
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Holomorphic strips versus holomorphic disks
What are the advantages of using holomorphic strips in Lagrangian intersection theory instead of using holomorphic disks?
I understand that for the analysis of strips one often needs to choose ...
3
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1
answer
129
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Holomorphic/Symplectic embedding of Riemann surfaces
Let $\Sigma_g$ denote a Riemann surface and let $X$ denote the complex surface $\Sigma_g \times \Sigma_g$. Then can there exist holomorphic embeddings of $\Sigma_l$ into $X$ for $l < g$?
What about ...
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symplectic gromov witten invariants of weighted projective space
Does anyone know if the symplectic (genus zero) GW invariants have been computed for weighted projective spaces? It seems there are already algebraic computations https://arxiv.org/abs/math/0608481
Is ...
- 113
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180
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Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?
Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...
- 21
4
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2
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448
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Dismissing pseudoholomorphic curves in embedded contact homology
In the papers
The periodic Floer homology of a Dehn twist,
Rounding corners of polygons and the embedded contact homology of $T^3$,
and Combinatorial embedded contact homology for toric contact ...
- 191
4
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0
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385
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Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology
I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
- 2,018
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0
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89
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Estimate for the diameter of the image of a holomorphic disk by the area of the holomorphic disk
Let $(M, J, g)$ be a compact almost complex manifold with a Riemannian metric $g$ that preserves the almost complex structure $J$. I want to prove that a holomorphic disk $u: D^2\to M$ of a small area ...
- 31
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115
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Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1
Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
- 113
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327
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Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold
Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
- 1,761
6
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172
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Computing Gromov-Witten invariant of $4$ lines in $\mathbb{C}P^3$
I'm trying to understand what the number of genus 0 curves through four lines in $\mathbb{C}P^3$ is i.e $Gr_{0,4}^{\mathbb{C}P^3, L}(PD(L),PD(L),PD(L),PD(L))$ where $L$ is the class of a line $\mathbb{...
- 755
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Is the forgetful map a submersion away from the nodal points?
Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow ...
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1
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161
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Extension of a holomorphic curve in $B^4$ to one in $\mathbb{C}P^2$
Let $B^4$ be the closed unit ball in $\mathbb{C}^2$ and $J$ an almost complex structure sufficiently closed to the standard complex structure on $\mathbb{C}^2$ in the $C^0$-topology. Let $u \colon S \...
- 369
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102
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Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries
Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
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139
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Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?
let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a ...
- 113
13
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1
answer
506
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How not to use J-holomorphic curves [closed]
The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
3
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71
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A clarification on why the injectivity radius is involved in Lemma 10.7 of Compactness results in Symplectic Field Theory by B.-E.-H.-W.-Z
I'm trying to understand why in the following lemma (Lemma 10.7 of [BEHWZ]), the upper bound on the $L^{\infty}$-norm of the differential is given in terms of the injective radius w.r.t to a specific ...
- 2,018
5
votes
1
answer
408
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Descent of vector bundle along branched cover of curve
Suppose $\pi:C'\to C$ is a branched cover of compact Riemann surfaces such that the associated extension of function fields is Galois with group $G$ -- so that $\pi$ presents $C$ as the quotient $C'$ ...
- 1,761
4
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1
answer
194
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Pseudo-holomorphic disk which is constant along boundary
Let $(M,J,\omega)$ be a symplectic manifold with a compatible almost complex structure, $D$ be the closed unit disk in $\mathbb{C}$, and $u:(D,i)\to (M,J)$ be a $(J,i)$-holomorphic map.
Question: ...
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1
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714
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Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends
I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in The Symplectic Floer Homology of a Dehn Twist.
I’ll try to summarize to the best ...
- 2,018
2
votes
1
answer
250
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Clarification on the ”neck stretching” applied to the base space of a Lefschetz fibration
I’m asking this question because I’d like to understand better the neck-stretching argument in symplectic geometry and what kind of conclusions one might get out of it in my setting.
Assume that I’ve ...
- 2,018
3
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1
answer
253
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Compactly supported symplectomorphisms of $D^2$
I'm trying to understand a more basic version about Gromov's theorem about the compactly supported symplectomorphisms of $D^2 \times D^2$ being contractible.
Consider the dimensional disk $D^2 \...
- 755
4
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1
answer
383
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Every symplectic submanifold is J-holomorphic
I am trying to show that every symplectic submanifold $N$ of a 2n- dimensional symplectic manifold $(M,\omega)$ is J-holomorphic for some compatible almost complex structure $J$.
The way I am ...
- 755
1
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0
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74
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Invariance under tame almost complex structure of the fibre tangent space of the symplectic normal bundle
I am trying to understand the construction of symplectic inflation and I am stuck in the following point.
Suppose we have a 4 dimensional symplectic manifold $(M, \omega)$. Also suppose that $N \...
- 755
7
votes
1
answer
369
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Strip breaking phenomenon in the Gromov compactification of Moduli space of Pseudoholomorphic curves
As the title suggests, I want to understand the strip breaking phenomenon that happens when I Gromov-compactify the moduli space of pseudoholomorphic curves from the holomorphic strip $\Bbb R \times [...
- 2,018
1
vote
1
answer
394
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Floer equation and Cauchy Riemann equation
Consider a symplectic manifold $(M,\omega)$ with the property that $\pi_2(M) = 0$. Given a time dependent hamiltonian $H_t$ on $M$, and a $\omega$-compatible almost complex structure J on M, we may ...
- 755
4
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0
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110
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Pairs of J-holomorphic curves
Let $(M, \omega)$ be a symplectic 4-manifold and let $A$ and $B$ be symplectic submanifolds on M such that $A \cap B = p \in M$.Under what conditions can I find a $\omega$-compatible almost complex ...
- 755
6
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1
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741
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Intuition about bubbling off a ghost bubble
I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that ...
- 2,018
4
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0
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127
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Diffeomorphism of $ \mathbb{C}P^2 \# ~\overline{\mathbb{C}P^2}$
I am currently reading Dusa McDuff's paper "Blow ups and symplectic embedding in dimension 4" and had a few questions regarding the paper.
In the paper McDuff uses the following notation. $X = \...
- 755
4
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0
answers
193
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Unobstructedness of nodal holomorphic curve in symplectic manifold
Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). ...
- 1,761
9
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1
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740
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DGLA controlling deformation of holomorphic curves
Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = ...
- 1,761
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1
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162
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Diameter of pseudoholomorphic curves
Fix an almost-complex structure $J$ on $\mathbb{R}^{2n}.$ Let $u: (D^2, i) \to (\mathbb{R}^{2n}, J)$ be a $J$-holomorphic disk.
My question: can one prove an a-priori bound on the diameter of $u$ (...
- 937
6
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1
answer
271
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More pseudoholomorphic curves than complex valued functions
A lecture I heard had a remark - "There is a rich class of pseudohoplomorphic curves to a symplectic manifold with an almost complex structure (tamed by the symplectic structure). On the other hand, ...
4
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1
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How to understand geometrically, the count of pseudoholomorphic discs by (multi)section perturbation of the kuranish structure on the moduli space?
When defining the $A_\infty$ algebra of a Lagrangian (as done in the book by FOOO) it is done by "counting" (integrating over the moduli space or over the fiber of evaluation map) pseudoholomorphic ...
- 1,893
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2
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444
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Fredholm property about $L^p$-extension $(p\neq 2)$ of differential operators
The following is a well-known result for elliptic operators.
Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact ...
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9
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1
answer
905
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Deligne Mumford Compactification of Moduli Space Of Annuli
I am reading Abouzaid's paper "A geometric criterion for generating the Fukaya category" (https://arxiv.org/abs/1001.4593), and it is claimed there, without proof, in section C.4 in the appendix (pp....
- 1,893
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96
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Can one choose a sufficiently generic path of a.c.s such that only "codimension 1" bubbling occurs?
Consider a symplectic manifold $(M,\omega)$ of dimension $2n$ (closed or open with bounded geometry). Let $L\subset M$ be a compact Lagrangian submanifold (not necessarily connected).
Consider two ...
- 1,893
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0
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97
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A genericity argument on family of disconnected holomorphic curves
Let $(W, \lambda)$ be an exact cobordism from $(M_+, \lambda_+)$ to $(M_-, \lambda_-)$ and $\overline{W}$ be the usual symplectic completion of $W$. Let $\mathcal{M}(J)$ be the module space of all ...
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539
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Most general maximum principle for non-integrable almost complex structures
Let $H\subseteq\mathbb C^n$ be a smooth co-oriented real codimension one hypersurface. If $H$ is weakly pseudo-convex, then holomorphic maps $u:\Delta\to\mathbb C^n$ ($\Delta$ denotes the unit disk) ...
- 18.7k
4
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91
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Topology of a convergent sequence of stable maps on a symplectic manifold
Let $(M,\omega)$ be a compact symplectic manifold. Let $J$ be a compatible almost complex structure. Let $g$ be the Riemannian metric corresponding to $\omega,J$.
Let $f_\nu\colon C_\nu\to M$ be a ...
- 21.8k
4
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0
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199
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infinite-dimensional transversality theorem and its application on the universal moduli space of pseudo-holomorphic curves
We would like to discuss Proposition 3.2.1 in McDuff&Salamon's book "J-holomorphic Curves and Symplectic Topology(Second Endition)".
Let me first remind you some background. Let $\Sigma$ be a ...
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1
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In $(\mathbb{R}^4,\omega_{std})$ is positive symplectic area enough to guarantee a pseudoholomorphic disc representative?
I will present my question in the context that I encountered it, although I believe it probably applies in general context.
Consider $\mathbb{R}^4 \cong \mathbb{C}^2$ with the standard symplectic form ...
- 1,893
1
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0
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147
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Shape of the bubbling limit of holomorphic discs
I will present my question in the specifics I encountered it, so maybe some of the details are irrelevant for the desired conclusion.
Consider $(S^2\times S^2,\omega_{std})$ the product of two ...
- 1,893
22
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1
answer
909
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Pseudo-holomorphic curves in the six-sphere
Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...
- 18.7k