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### MINIMAL SURFACES

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. I briefly know that we can study minimal surfaces by three different directions, which are complex analysis, PDE and geometric measure theory. I wish to study about how complex analysis can be used to study minimal surfaces.

Is there any good introductory book on this topic? Is Riemannian Geometry required to study minimal surfaces? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Good introductory book on minimal surfaces.

Ask Question. Asked 1 year, 4 months ago. Active 1 year, 4 months ago. Viewed times. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Responding to the Lavender Letter and commitments moving forward. Related Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.In mathematicsthe Enriques—Kodaira classification is a classification of compact complex surfaces into ten classes.

For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.

Max Noether began the systematic study of algebraic surfaces, and Guido Castelnuovo proved important parts of the classification.

For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like which for class VII depends on the global spherical shell conjecturestill unproved in For surfaces of general type not much is known about their explicit classification, though many examples have been found. Oscar Zariski constructed some surfaces in positive characteristic that are unirational but not rational, derived from inseparable extensions Zariski surfaces.

The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups. The basic ones are the plurigenera and the Hodge numbers defined as follows:. There are many invariants that at least for complex surfaces can be written as linear combinations of the Hodge numbers, as follows:.

There are further invariants of compact complex surfaces that are not used so much in the classification. Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces.

**Mysteries of Minimal Surfaces**

Given any point on a surface, we can form a new surface by blowing up this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface X is called minimal if it cannot be obtained from another non-singular surface by blowing up a point.

In the more modern terminology of the minimal model programa smooth projective surface X would be called minimal if its canonical line bundle K X is nef. A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative. Every surface X is birational to a minimal non-singular surface, and this minimal non-singular surface is unique if X has Kodaira dimension at least 0 or is not algebraic. So to classify all compact complex surfaces up to birational isomorphism it is more or less enough to classify the minimal non-singular ones.

For non-algebraic surfaces Kodaira found an extra class of surfaces, called type VII, which are still not well understood.

## Minimal surface

Rational surface means surface birational to the complex projective plane P 2. These are all algebraic. Many of these examples are non-minimal. Ruled surfaces of genus g have a smooth morphism to a curve of genus g whose fibers are lines P 1.

They are all algebraic.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. What did I do wrong? Can someone help me please? I think you are a little bit confused about the harmonic characterization of conformally immersed minimal surfaces.

I would advise going back to look at the proof of this characterization for clarification, and thinking about geometrically what it means to be conformal preserve angles. I think where you have been misled is in thinking of this as a polar parametrization and using the so-called "polar form of the laplacian.

They are just abstract coordinates. Sign up to join this community.

Mtd transaxle greaseThe best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Catenoid is a minimal surface Ask Question. Asked 2 years, 4 months ago. Active 2 years, 4 months ago. Viewed times. Thanks in advance. Jean-Claude Arbaut That characterization you're trying to use is wrong.

See the Harmonic definition here: en. The requirement we needed was that the surface is conformity parameterized. Active Oldest Votes. Badam Baplan Badam Baplan 6, 11 11 silver badges 27 27 bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Responding to the Lavender Letter and commitments moving forward.

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### Variational principles and global analysis

Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.The great mathematicians of Classical times were very interested in variational problems.

An example is the famous problem of the brachistochrone: find the shape of a curve with given start and end points along which a body will fall in the shortest possible time. The answer is part of an upside-down cycloid, where a cycloid is the path traced by a point on the rim of a rolling circle. More important for the purposes of this article is the nature of the problem: from among a class of curves, select the one that minimizes some quantity.

Variational problems can be put into Banach space language too. The space of curves is the Banach space, the quantity to be minimized is some functional a function with functions, rather than simply numbers, as input defined on the Banach space, and the methods of analysis can be used to determine the minimum.

This approach can be generalized even further, leading to what is now called global analysis. Global analysis has many applications to mathematical physics. Euler and the French mathematician Pierre-Louis Moreau de Maupertuis discovered that the whole of Newtonian mechanics can be restated in terms of a variational principle: mechanical systems move in a manner that minimizes or, more technically, extremizes a functional known as action.

The French mathematician Pierre de Fermat stated a similar principle for optics, known as the principle of least time: light rays follow paths that minimize the total time of travel. Later the Irish mathematician William Rowan Hamilton found a unified theory that includes both optics and mechanics under the general notion of a Hamiltonian system—nowadays subsumed into a yet more general and abstract theory known as symplectic geometry.

An especially fascinating area of global analysis concerns the Plateau problem. The blind Belgian physicist Joseph Plateau using an assistant as his eyes spent many years observing the form of soap films and bubbles.

He found that if a wire frame in the form of some curve is dipped in a soap solution, then the film forms beautiful curved surfaces. They are called minimal surfaces because they have minimal area subject to spanning the curve.

Their surface tension is proportional to their area, and their energy is proportional to surface tension, so they are actually energy-minimizing films.

For example, a soap bubble is spherical because a sphere has the smallest surface area, subject to enclosing a given volume of air. The mathematics of minimal surfaces is an exciting area of current research with many attractive unsolved problems and conjectures. One of the major triumphs of global analysis occurred in when the American mathematicians Jean Taylor and Frederick Almgren obtained the mathematical derivation of the Plateau conjecture, which states that, when several soap films join together for example, when several bubbles meet each other along common interfacesthe angles at which the films meet are either degrees for three films or approximately degrees for four films.

Plateau had conjectured this from his experiments. One philosophical feature of traditional analysis, which worries mathematicians whose outlook is especially concrete, is that many basic theorems assert the existence of various numbers or functions but do not specify what those numbers or functions are.

For instance, the completeness property of the real numbers indicates that every Cauchy sequence converges but not what it converges to. A school of analysis initiated by the American mathematician Errett Bishop has developed a new framework for analysis in which no object can be deemed to exist unless a specific rule is given for constructing it.

This school is known as constructive analysis, and its devotees have shown that it is just as rich in structure as traditional analysis and that most of the traditional theorems have analogs within the constructive framework.In mathematicsa minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature see definitions below. The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint.

Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap filmwhich is a minimal surface whose boundary is the wire frame.

However, the term is used for more general surfaces that may self-intersect or do not have constraints.

For a given constraint there may also exist several minimal surfaces with different areas for example, see minimal surface of revolution : the standard definitions only relate to a local optimumnot a global optimum. Minimal surfaces can be defined in several equivalent ways in R 3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometrycalculus of variationspotential theorycomplex analysis and mathematical physics.

This property is local: there might exist other surfaces that minimize area better with the same global boundary. This definition makes minimal surfaces a 2-dimensional analogue to geodesics. By the Young—Laplace equation the curvature of a soap film is proportional to the difference in pressure between the sides: if it is zero, the membrane has zero mean curvature. Spherical bubbles are not minimal surfaces as per this definition: while they minimize total area subject to a constraint on internal volume, they have a positive pressure.

A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. The partial differential equation in this definition was originally found in by Lagrange[2] and Jean Baptiste Meusnier discovered in that it implied a vanishing mean curvature.

Blue harlequin great dane priceThis definition ties minimal surfaces to harmonic functions and potential theory. A direct implication of this definition and the maximum principle for harmonic functions is that there are no compact complete minimal surfaces in R 3. This definition uses that the mean curvature is half of the trace of the shape operatorwhich is linked to the derivatives of the Gauss map.

If the projected Gauss map obeys the Cauchy—Riemann equations then either the trace vanishes or every point of M is umbilicin which case it is a piece of a sphere.

The local least area and variational definitions allow extending minimal surfaces to other Riemannian manifolds than R 3. He derived the Euler—Lagrange equation for the solution. He did not succeed in finding any solution beyond the plane. In Jean Baptiste Marie Meusnier discovered that the helicoid and catenoid satisfy the equation and that the differential expression corresponds to twice the mean curvature of the surface, concluding that surfaces with zero mean curvature are area-minimizing.

Gaspard Monge and Legendre in derived representation formulas for the solution surfaces. While these were successfully used by Heinrich Scherk in to derive his surfacesthey were generally regarded as practically unusable. The "first golden age" of minimal surfaces began. Schwarz found the solution of the Plateau problem for a regular quadrilateral in and for a general quadrilateral in allowing the construction of his periodic surface families using complex methods.

Weierstrass and Enneper developed more useful representation formulasfirmly linking minimal surfaces to complex analysis and harmonic functions.

Between and minimal surface theory revived, now mainly aimed at nonparametric minimal surfaces.

Bernstein's problem and Robert Osserman 's work on complete minimal surfaces of finite total curvature were also important. Another revival began in the s. One cause was the discovery in by Celso Costa of a surface that disproved the conjecture that the plane, the catenoid, and the helicoid are the only complete embedded minimal surfaces in R 3 of finite topological type.

This not only stimulated new work on using the old parametric methods, but also demonstrated the importance of computer graphics to visualise the studied surfaces and numerical methods to solve the "period problem" when using the conjugate surface method to determine surface patches that can be assembled into a larger symmetric surface, certain parameters need to be numerically matched to produce an embedded surface.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. What did I do wrong? Can someone help me please? I think you are a little bit confused about the harmonic characterization of conformally immersed minimal surfaces.

I would advise going back to look at the proof of this characterization for clarification, and thinking about geometrically what it means to be conformal preserve angles. I think where you have been misled is in thinking of this as a polar parametrization and using the so-called "polar form of the laplacian.

They are just abstract coordinates. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Catenoid is a minimal surface Ask Question. Asked 2 years, 4 months ago. Active 2 years, 4 months ago. Viewed times. Thanks in advance. Jean-Claude Arbaut That characterization you're trying to use is wrong. See the Harmonic definition here: en. The requirement we needed was that the surface is conformity parameterized.

Active Oldest Votes. Badam Baplan Badam Baplan 6, 11 11 silver badges 27 27 bronze badges. Sign up or log in Sign up using Google.

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