The curves-and-surfaces tag has no wiki summary.

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**1**answer

100 views

### Reduction of self-intersections without reducing the geometric intersection

Let $F$ be a hyperbolic surface. Given a closed curve $a$, let $\bar{a}$ denotes the free homotopy class of $a$. Let $i(\bar{a},\bar{b})$ denotes the geometric intersection number and $i(\bar{a})$ ...

**1**

vote

**1**answer

126 views

### Image of any curve can be parametrized without zero derivative?

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a $C^{1} ([a,b])$ injective application. Is it true that there is another continuous parametrization $\rho:[c,d]\to\mathbb{R}^2$ such that the following two sets ...

**0**

votes

**1**answer

70 views

### Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...

**3**

votes

**1**answer

235 views

### On equations defining space curves

I am reading a text by Prof. Szpiro Tata lectures on equations defining space curves. In the proof of Proposition $1.2$ on page $12$ he gives explicit description of the defining equations of a local ...

**4**

votes

**1**answer

215 views

### Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture:
Every object naturally associated to a surface S and having
a sufficiently rich structure has $Mod(S)$ as its groups of ...

**0**

votes

**0**answers

37 views

### Unitary derivative and countable set

Let $\mathbf{r}:I\to\mathbb{R}^2$, where $I\subseteq\mathbb{R}$ is an open interval, be a continuous function that is not constant on any subinterval $J\subseteq I$ such that at each point $t\in I$ ...

**1**

vote

**1**answer

122 views

### How to find isothermal coordinates equivalent to circles in far limit?

I am trying to find the most general rotational coordinate systems for Euclidean 3-space, with the following two defining characteristics: 1) being equivalent to spherical coordinates in the limit of ...

**0**

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**0**answers

82 views

### Positive curvature of the boundary away from a point implies regularity?

In a paper I'm refereeing, the authors make use of the following geometric fact:
Let $U$ be an open subset of $\mathbb{R}^2$. If there is a point $p\in \partial U$ so that $\partial U \backslash p$ ...

**3**

votes

**0**answers

188 views

### Hypersurface with singularities

I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).
It was said ...

**2**

votes

**0**answers

64 views

### Topological/numerical constraints for the existence of more than one pencil

A famous theorem of Castelnuovo and de Franchis tells us that for $S$ a smooth projective complex algebraic surface that for $b \geq 2$, pencils $f : S \to B$ of genus $b :=g(B)$ are in bijective ...

**0**

votes

**2**answers

88 views

### Planar curves identical to their inverses

Is the right strophoid
the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin)
is identical to $C$?
...

**1**

vote

**1**answer

106 views

### Characterization of $d$-gonal curves on a K3 surface

Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).
I am wondering if there exist characterizations of ...

**4**

votes

**0**answers

196 views

### Are there Zoll pancakes?

How flat (flat in pancake-style, not in curvature 0-style), in some extrinsic intuitive measure, can a Zoll surface of revolution (embedded in Euclidean three-space) be?
I don't want to impose a ...

**0**

votes

**0**answers

41 views

### Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...

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**0**answers

103 views

### Is there a unique solution? [closed]

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...

**1**

vote

**1**answer

132 views

### Another type of derivative, and the associated primitive

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Find all continuous functions $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so that:
$
...

**9**

votes

**0**answers

81 views

### Area of the minimal surface of a non-planar quadrilateral in 3d

Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of ...

**1**

vote

**0**answers

55 views

### General reparameterization of a b-spline

Say I have a bspline function (or curve) of order $k_1$, defined over some knot vector
$\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$
Do you know of a process of finding ...

**0**

votes

**0**answers

69 views

### Intersection points of closed curves inscribed in a convex polygon

Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...

**3**

votes

**5**answers

470 views

### Variation of curvature with respect to immersion?

Let $M$ be a smooth surface and let $f: M \to \mathbb{R}^3$ be a family of immersions given by
$$ f(t) = f_0 + tuN_0, $$
where $f_0$ is some initial immersion, $N_0$ is the associated Gauss map, and ...

**4**

votes

**1**answer

138 views

### Angle between geodesics in hyperbolic surface

Let $F$ be an oriented surface of finite type with $\chi(F)<0$. Let $\gamma_1$ and $\gamma_2$ are two oriented closed curves which intersect transversally in double points. Given a hyperbolic ...

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**0**answers

211 views

### “Spreading out” locally free sheaves

Let $Y$ be a regular surface flat, projective over $R$, where $R$ is complete DVR. Let $X$ be the generic fiber and $F$ be a locally free sheaf on $X$. We know that if the rank of $F$ is $1$ then we ...

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**0**answers

282 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...

**2**

votes

**1**answer

124 views

### Intersection of closed geodesics in hyperbolic surface

This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed ...

**5**

votes

**2**answers

409 views

### Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?

There is no surface in $ R^3 $ that can represent the complete hyperbolic plane (Hilberts theorem) so we always have to do with a surface that is not completely equivalent, has a cusp somewhere, but ...

**3**

votes

**1**answer

81 views

### How to approximate volumes of m-dimensional manifolds with m-dimensional polyhedra

The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter ...

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**0**answers

211 views

### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

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votes

**1**answer

132 views

### A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...

**1**

vote

**1**answer

71 views

### On Severi's definition of the complementary correspondence

In Weil's short note entitled "On the Riemann hypothesis in function-fields"
he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where ...

**3**

votes

**2**answers

219 views

### Effective cone of C x C where C is a Fermat curve

A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of ...

**2**

votes

**1**answer

204 views

### The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math.
Let's consider two examples of curves:
1)The curve defined by the ...

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**0**answers

135 views

### Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...

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votes

**1**answer

131 views

### Curvature and Failure to return to starting point

Assume I have a geodesic polygon $P$ in a Riemannian manifold $M$ that is given by the image of a piecewise geodesic closed curve $\gamma(t)$ (parametrized by arclength), with vertices $x_i = ...

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**0**answers

39 views

### Connectivity and contarctibility of complexes associated to curves and arcs

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to ...

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**1**answer

114 views

### A doubt from “Geometry of the complex of curves II: Hierarchical structure” by Masur and Minsky

In the paper "Geometry of the complex of curves II: Hierarchical structure" (Paper) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain ...

**1**

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**1**answer

98 views

### Invariant of isotopy of curves in a surface.

Suppose $S_g$ is a sorface of genus $g>1$. Let $\gamma_1$ and $\gamma_2$ be two simple closed curves containing points $p_1, p_2$. Suppose $\gamma_1$ and $\gamma_2$ are isotopic. Now there can be ...

**2**

votes

**1**answer

133 views

### Quotient of product of curves

Let $C_1,C_2$ be smooth, projective curves of genera $g_1,g_2 \geq 2$. Assume that a group $G$ of order $(g_1 - 1)(g_2 - 1)$ acts on $C_1$ and $C_2$ such that $C_1/G \cong \mathbb{P}^1$ and $C_2/G ...

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**0**answers

48 views

### A question on the curvature of smooth and embedded curves in a plane

I am from China. This is my first question in this website.
Given a family of embedded, smooth and closed curves $\{X_i(s)\}_{i=1}^\infty$ in a plane, let us denote by $L_i$ the perimeter and ...

**2**

votes

**1**answer

196 views

### Surface curves equidistant from a simple closed geodesic

Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is ...

**4**

votes

**3**answers

315 views

### Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$

Let $Q\cong\mathbb{P^{1}_{1}}\times\mathbb{P^{1}_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S_2\times Q\rightarrow Q,\; ...

**14**

votes

**2**answers

734 views

### How useful/pervasive are differential forms in surface theory?

Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an ...

**2**

votes

**2**answers

193 views

### Curve of 3-secant lines

Let $C\subset\mathbb{P}^{3}$ be a smooth, non-degenerate curve over an algebraically closed field of characteristic zero. Let $d$ be the degree of $C$ and $g$ be its genus.
Consider the variety ...

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**1**answer

192 views

### Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'

Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...

**1**

vote

**1**answer

198 views

### On the dualizing sheaf of a curve

Let $X$ be a smooth projective surface in $\mathbb{P}^n$ and $C$ be an effective curve. I know that the dualizing sheaf, $\omega_C$ of $C$ is ...

**2**

votes

**1**answer

568 views

### base points of multiplicity $>1$

Let $S$ be a smooth projective surface (I am mostly intrested in the case when $S$ is a product of curves, say $S=\mathbb{P}^1 \times \mathbb{P}^1$ but probably this is not important).
Consider a ...

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**0**answers

184 views

### Embedding a projective curve in a smooth surface (using a Bertini theorem)

Let $C$ be a reduced (not irreducible) projective curve of degree $d$ such that $C$ contains at most double points. By a result due to Kleiman and Altman, we know that there exists a smooth surface ...

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**0**answers

182 views

### Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...

**14**

votes

**4**answers

1k views

### What is the analog of the “Fundamental Theorem of Space Curves,” for surfaces, and beyond?

The "Fundamental Theorem of Space Curves"
(Wikipedia link; MathWorld link)
states that there is a unique (up to congruence)
curve in space that simultaneously realizes
given continuous curvature ...

**3**

votes

**2**answers

293 views

### Realizing homology classes on surfaces with boundary by simple closed curves

Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...

**1**

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**1**answer

117 views

### Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...