mathematics

Maximal surface

Maximal surface is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Maximal surface rather than just read about it. In short: In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold (M, g), a maximal surface is a spacelike submanifold of M whose mean curvature is zero.

Key takeaways

Maximal surface belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Maximal surface to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Maximal surface from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of differential geometry, a maximal surface is a certain kind of submanifold of a Lorentzian manifold. Precisely, given a Lorentzian manifold (M, g), a maximal surface is a spacelike submanifold of M whose mean curvature is zero. As such, maximal surfaces in Lorentzian geometry are directly analogous to minimal surfaces in Riemannian geometry. The difference in terminology between the two settings has to do with the fact that small regions in maximal surfaces are local maximizers of the area functional, while small regions in minimal surfaces are local minimizers of the area functional. In 1976, Shiu-Yuen Cheng and Shing-Tung Yau resolved the Bernstein problem for maximal surfaces of Minkowski space which are properly embedded, showing that any such hypersurface is a plane. This was part of the body of work for which Yau was awarded the Fields Medal in 1982. The Bernstein problem was originally posed by Eugenio Calabi in 1970, who proved some special cases of the result. Simple examples show that there are a number of hypersurfaces of Minkowski space of zero mean curvature which fail to be spacelike. By an extension of Cheng and Yau's methods, Kazuo Akutagawa considered the case of spacelike hypersurfaces of constant mean curvature in Lorentzian manifolds of positive constant curvature, such as de Sitter space. Luis Alías, Alfonso Romero, and Miguel Sánchez proved a version of Cheng and Yau's result, replacing Minkowski space by the warped product of a closed Riemannian manifold with an interval. As a problem of partial differential equations, Robert Bartnik and Leon Simon studied the boundary-value problem for maximal surfaces in Minkowski space. The general existence of maximal hypersurfaces in asymptotically flat Lorentzian manifolds, due to Bartnik, is significant in Demetrios Christodoulou and Sergiu Klainerman's renowned proof of the nonlinear stability of Minkowski space under the Einstein field equations. They use a maximal slicing of a general spacetime; the same approach is common in numerical relativity.

References

Worked examples

Example 1 — a first encounter with Maximal surface

Start with the simplest possible case. Write down what Maximal surface claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Maximal surface before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Maximal surface ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Maximal surface

In research: Maximal surface appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Maximal surface in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Maximal surface is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Lorentzian manifolds, so understanding it makes those chapters shorter.
In everyday life: Look for Maximal surface outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Maximal surface in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Maximal surface means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Maximal surface out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Maximal surface in simple terms?

Why does Maximal surface matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Maximal surface?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Maximal surface.