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I've combined two answers into one.

First answer: some results on gradient Ricci solitons including some history. I like very much the answer Otis Chodosh gave. To this I would add the following: Besides the physics literature (e.g., Friedan), the notion of gradient Ricci soliton first appeared in Hamilton's Ricci flow on surfaces paper to deal with the $\chi >0$ case. The idea being that doing geometric analysis, such as in Ricci flow, one often cannot differentiate between smooth manifolds and orbifolds with isolated singularities. When a closed 2-orbifold $M^2$ has $\chi >0$, it may be a bad orbifold (not globally covered by a smooth surface) and in this case there does not exist a constant curvature metric on $M$ (this is not the case when $\chi \leq 0$). For bad 2-orbifolds, the normalized Ricci flow evolves any metric to a shrinking gradient Ricci soliton metric asymptotically; this was proved for initial metric with positive curvature by L.-F. Wu and in the general case by L.-F. Wu and myself.

This is just a selection of results; I'm sure I inadvertently omitted some I wanted to mention. By now, there are many more important papers in the field of Ricci solitons, perhaps too many to mention (I've omitted the Kaehler case; see for example Feldman--Ilmanen--Knopf for constructions with symmetry as Otis Chodosh mentioned as well as Dancer--Wang, Bo Yang, etal.; and the work of Xiaohua Zhu and coauthors in relation to Kaehler--Ricci flow). One just can search arXiv to see most of the recent papers.

Second answer: some statements and questions. This is a supplement to my previous post. Consider a shrinker model $(M^{n},g,f)$, i.e., a complete shrinking gradient Ricci soliton which is a singularity model for some Ricci flow $\tilde{g}(t)$ on a closed manifold $N^{n}$. If $M$ is compact, then it is diffeomorphic to $N$. So one may focus on the case where $M$ is noncompact. Actually, any compact singularity model must be a shrinker by Z.-L. Zhang.

By Perelman's no local collapsing theorem, there exists $\kappa>0$ such that $M$ is $\kappa$-noncollapsed below all scales, i.e., if $R\leq r^{-2}$ in $B_{p}(r)$, then $\operatorname{Vol}B_{p}(r)\geq\kappa r^{n}$. (There is no restriction on the scale since singularities arise from rescaling factors tending to $\infty$.) This implies that steady models have at least linear volume growth; Munteanu and J.-P. Wang have proved this without the model assumption. The following applies to all complete nonflat shrinkers (not necessarily models).

  1. $R>0$ by B.-L. Chen; in fact, $R(x)\geq c(d(x,O)+1)^{-2}$ by L. Ni and B. Wilking and joint with P. Lu and B. Yang. So any asymptotic cone must be nonflat.

  2. $\operatorname{Vol}B_{p}(r)\leq C_{0}r^{n}$ by H.-D. Cao and D.-T. Zhou and Munteanu.

  3. The asymptotic volume ratio $\operatorname{AVR}\doteqdot\lim_{r\rightarrow \infty}\frac{\operatorname{Vol}B_{p}(r)}{r^{n}}$ exists. We have $\operatorname{AVR}>0$ iff $\int^{\infty}\frac{dr}{r\operatorname{Vol} B_{p}(r)}\int_{B_{p}(r)}Rd\mu<\infty$ (intuitively, the average scalar curvature on balls decays at least a little) by joint with P. Lu and B. Yang, based on numerous previous works.

Chen's result is a localization. The other results depend on the following fundamental result of Cao and Zhou: Assume the normalizations $\operatorname{Ric}+\nabla^{2}f=\frac{1}{2}g$ and $R+|\nabla f|^{2}=f$; then $$ \frac{1}{4}(d(x,O)-C_{1})_{+}^{2}\leq f\left( x\right) \leq\frac{1} {4}(d(x,O)+C_{2})^{2}. $$ By Haslhofer and Mueller, if $O$ is a minimum point of $f$, then we may take $C_{1}=5n$ and $C_{2}=\sqrt{2n}$; consequently $C_{0}$ in (2) depends only on $n$. (Could $C_{0}$ be the Euclidean constant?)

One may make an analogy between gradient Ricci solitons (GRS) and Einstein metrics and one might even say that GRS are harder to study than Einstein metrics since they are more general. However, the Einstein special case of shrinkers implies compactness by Myers' theorem. For manifolds with positive (or nonnegative) sectional curvature, Robert Greene wrote: "Thus some tension arises between the necessary existence of rays and the curvature's nonnegativity." Perhaps, similarly, there is some tension between shrinking and noncompactness.

Indeed, for noncompact shrinkers, Cao and Zhou's potential function estimate seems to indicate some sort of rigidity. Furthermore, in arXiv:1307.4746 Brett Kotschwar and Lu Wang prove that any two complete shrinkers with a pair of ends that are asymptotic to the same regular cone must have isometric universal covers.

For a shrinker, must the asymptotic cone always exist, be unique, and be regular? However, one does not know any good bounds on the curvature tensor of a shrinker, let alone quadratic curvature decay. For instance, finite topological type is unknown even though it would follow from an estimate of the form: $R\leq\frac{1}{4+\varepsilon}(d(x,O)+C_{3})^{2}$, where $\varepsilon>0$.

Added December 13, 2013: A note on the etymology of the word "soliton" in the context of Ricci flow. It first appears in Richard Hamilton's "The Ricci flow on surfaces" paper, which discusses the $2$-dimensional rotationally symmetric case, including the cigar soliton and gradient Ricci solitons on $2$-orbifolds. Hamilton likes to joke that Shing-Tung Yau, his colleague at UCSD in the early 80's, was applying for a grant to study a topic related to solitons and hence the coinage of the word soliton. Under the Ricci flow in the space of metrics modulo diffeomorphisms acting by pullback, the cigar soliton is a fixed point. It evolves by "translating" by the pullback by homotheties of $\mathbb{R} ^{2}$ (remove a point to get the cylinder), but it may still be a stretch to say that it is a wave of translation.

I like very much the answer Otis Chodosh gave. To this I would add the following: Besides the physics literature (e.g., Friedan), the notion of gradient Ricci soliton first appeared in Hamilton's Ricci flow on surfaces paper to deal with the $\chi >0$ case. The idea being that doing geometric analysis, such as in Ricci flow, one often cannot differentiate between smooth manifolds and orbifolds with isolated singularities. When a closed 2-orbifold $M^2$ has $\chi >0$, it may be a bad orbifold (not globally covered by a smooth surface) and in this case there does not exist a constant curvature metric on $M$ (this is not the case when $\chi \leq 0$). For bad 2-orbifolds, the normalized Ricci flow evolves any metric to a shrinking gradient Ricci soliton metric asymptotically; this was proved for initial metric with positive curvature by L.-F. Wu and in the general case by L.-F. Wu and myself.

This is just a selection of results; I'm sure I inadvertently omitted some I wanted to mention. By now, there are many more important papers in the field of Ricci solitons, perhaps too many to mention (I've omitted the Kaehler case; see for example Feldman--Ilmanen--Knopf for constructions with symmetry as Otis Chodosh mentioned as well as Dancer--Wang, Bo Yang, etal.; and the work of Xiaohua Zhu and coauthors in relation to Kaehler--Ricci flow). One just can search arXiv to see most of the recent papers.

I've combined two answers into one.

First answer: some results on gradient Ricci solitons including some history. I like very much the answer Otis Chodosh gave. To this I would add the following: Besides the physics literature (e.g., Friedan), the notion of gradient Ricci soliton first appeared in Hamilton's Ricci flow on surfaces paper to deal with the $\chi >0$ case. The idea being that doing geometric analysis, such as in Ricci flow, one often cannot differentiate between smooth manifolds and orbifolds with isolated singularities. When a closed 2-orbifold $M^2$ has $\chi >0$, it may be a bad orbifold (not globally covered by a smooth surface) and in this case there does not exist a constant curvature metric on $M$ (this is not the case when $\chi \leq 0$). For bad 2-orbifolds, the normalized Ricci flow evolves any metric to a shrinking gradient Ricci soliton metric asymptotically; this was proved for initial metric with positive curvature by L.-F. Wu and in the general case by L.-F. Wu and myself.

This is just a selection of results; I'm sure I inadvertently omitted some I wanted to mention. By now, there are many more important papers in the field of Ricci solitons, perhaps too many to mention (I've omitted the Kaehler case; see for example Feldman--Ilmanen--Knopf for constructions with symmetry as Otis Chodosh mentioned as well as Dancer--Wang, Bo Yang, etal.; and the work of Xiaohua Zhu and coauthors in relation to Kaehler--Ricci flow). One just can search arXiv to see most of the recent papers.

Second answer: some statements and questions. This is a supplement to my previous post. Consider a shrinker model $(M^{n},g,f)$, i.e., a complete shrinking gradient Ricci soliton which is a singularity model for some Ricci flow $\tilde{g}(t)$ on a closed manifold $N^{n}$. If $M$ is compact, then it is diffeomorphic to $N$. So one may focus on the case where $M$ is noncompact. Actually, any compact singularity model must be a shrinker by Z.-L. Zhang.

By Perelman's no local collapsing theorem, there exists $\kappa>0$ such that $M$ is $\kappa$-noncollapsed below all scales, i.e., if $R\leq r^{-2}$ in $B_{p}(r)$, then $\operatorname{Vol}B_{p}(r)\geq\kappa r^{n}$. (There is no restriction on the scale since singularities arise from rescaling factors tending to $\infty$.) This implies that steady models have at least linear volume growth; Munteanu and J.-P. Wang have proved this without the model assumption. The following applies to all complete nonflat shrinkers (not necessarily models).

  1. $R>0$ by B.-L. Chen; in fact, $R(x)\geq c(d(x,O)+1)^{-2}$ by L. Ni and B. Wilking and joint with P. Lu and B. Yang. So any asymptotic cone must be nonflat.

  2. $\operatorname{Vol}B_{p}(r)\leq C_{0}r^{n}$ by H.-D. Cao and D.-T. Zhou and Munteanu.

  3. The asymptotic volume ratio $\operatorname{AVR}\doteqdot\lim_{r\rightarrow \infty}\frac{\operatorname{Vol}B_{p}(r)}{r^{n}}$ exists. We have $\operatorname{AVR}>0$ iff $\int^{\infty}\frac{dr}{r\operatorname{Vol} B_{p}(r)}\int_{B_{p}(r)}Rd\mu<\infty$ (intuitively, the average scalar curvature on balls decays at least a little) by joint with P. Lu and B. Yang, based on numerous previous works.

Chen's result is a localization. The other results depend on the following fundamental result of Cao and Zhou: Assume the normalizations $\operatorname{Ric}+\nabla^{2}f=\frac{1}{2}g$ and $R+|\nabla f|^{2}=f$; then $$ \frac{1}{4}(d(x,O)-C_{1})_{+}^{2}\leq f\left( x\right) \leq\frac{1} {4}(d(x,O)+C_{2})^{2}. $$ By Haslhofer and Mueller, if $O$ is a minimum point of $f$, then we may take $C_{1}=5n$ and $C_{2}=\sqrt{2n}$; consequently $C_{0}$ in (2) depends only on $n$. (Could $C_{0}$ be the Euclidean constant?)

One may make an analogy between gradient Ricci solitons (GRS) and Einstein metrics and one might even say that GRS are harder to study than Einstein metrics since they are more general. However, the Einstein special case of shrinkers implies compactness by Myers' theorem. For manifolds with positive (or nonnegative) sectional curvature, Robert Greene wrote: "Thus some tension arises between the necessary existence of rays and the curvature's nonnegativity." Perhaps, similarly, there is some tension between shrinking and noncompactness.

Indeed, for noncompact shrinkers, Cao and Zhou's potential function estimate seems to indicate some sort of rigidity. Furthermore, in arXiv:1307.4746 Brett Kotschwar and Lu Wang prove that any two complete shrinkers with a pair of ends that are asymptotic to the same regular cone must have isometric universal covers.

For a shrinker, must the asymptotic cone always exist, be unique, and be regular? However, one does not know any good bounds on the curvature tensor of a shrinker, let alone quadratic curvature decay. For instance, finite topological type is unknown even though it would follow from an estimate of the form: $R\leq\frac{1}{4+\varepsilon}(d(x,O)+C_{3})^{2}$, where $\varepsilon>0$.

Added December 13, 2013: A note on the etymology of the word "soliton" in the context of Ricci flow. It first appears in Richard Hamilton's "The Ricci flow on surfaces" paper, which discusses the $2$-dimensional rotationally symmetric case, including the cigar soliton and gradient Ricci solitons on $2$-orbifolds. Hamilton likes to joke that Shing-Tung Yau, his colleague at UCSD in the early 80's, was applying for a grant to study a topic related to solitons and hence the coinage of the word soliton. Under the Ricci flow in the space of metrics modulo diffeomorphisms acting by pullback, the cigar soliton is a fixed point. It evolves by "translating" by the pullback by homotheties of $\mathbb{R} ^{2}$ (remove a point to get the cylinder), but it may still be a stretch to say that it is a wave of translation.

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user41263
user41263

I like very much the answer Otis Chodosh gave. To this I would add the following: Besides the physics literature (e.g., Friedan), the notion of gradient Ricci soliton first appeared in Hamilton's Ricci flow on surfaces paper to deal with the $\chi >0$ case. The idea being that doing geometric analysis, such as in Ricci flow, one often cannot differentiate between smooth manifolds and orbifolds with isolated singularities. When a closed 2-orbifold $M^2$ has $\chi >0$, it may be a bad orbifold (not globally covered by a smooth surface) and in this case there does not exist a constant curvature metric on $M$ (this is not the case when $\chi \leq 0$). For bad 2-orbifolds, the normalized Ricci flow evolves any metric to a shrinking gradient Ricci soliton metric asymptotically; this was proved for initial metric with positive curvature by L.-F. Wu and in the general case by L.-F. Wu and myself.

Secondly, steady gradient Ricci solitons were necessarily obtained as Eternal solution in the paper by Hamilton by that name. Here the idea was to use his so-called matrix Harnack estimate (a differential Harnack estimate in the spirit of Li--Yau). Intuitively, the idea is that Hamilton's Harnack estimate represents some sort of nonnegative curvature (see his Formation of Singularities paper for this) and one applies his strong maximum principle for systems (after Weinberger in the PDE setting) to get a space-time splitting of the solution, which tells use an eternal solution (with some assumptions including $\operatorname{Rm}>0$) must be a steady gradient Ricci soliton.

For the early study of Ricci solitons, see the works of Tom Ivey (and of Robert Bryant in the rotationally symmetric case, as Otis Chodosh mentioned). The study of the geometry at infinity of gradient Ricci solitons was first done by Hamilton motivated by singularity analysis (see his Formation paper).

Perelman made great advances in the study of gradient Ricci solitons via his monotonicity formulae and other arguments. This was applied to his study of Ricci flow in all dimensions and also in dimension three, where notably he proved that there does not exist any (complete) noncompact shrinking gradient Ricci soliton with bounded positive curvature. Extensions of this result as well as other classification results were given by Ni--Wallach, Naber, Petersen--Wylie, H.D. Cao and coauthors, etal.

A fundamental paper in the study of the geometry of shrinkers is that by H.D. Cao and D.-T. Zhou. They proved a qualitatively sharp lower bound for the potential function in terms of the squared distance (the sharp upper bound is easy). Using this they proved that shrinkers have at most Euclidean volume growth under a weak technical condition. Later, using a result of B.L. Chen (ancient solutions must have nonnegative scalar curvature), Munteanu removed this condition; this result result was incorporated in the revised and published paper by Cao and Zhou. Regarding the behavior of the potential function, a related result was obtained by Fang, Man and Z.-L. Zhang.

Returning to steady gradient Ricci solitons, a breakthrough was made by Brendle who proved a claim/conjecture/stated theorem of Perelman on the uniqueness of the Bryant soliton in dimension 3 in the realm of nonflat $\kappa$-noncollapsed steadies, perhaps with bounded curvature. This paper has led to further developments on understanding symmetry of Ricci solitons.

The papers of Munteanu and Jiaping Wang apply geometric analysis and function theory techniques to the study of gradient Ricci solitons and, more generally, so-called Bakry--Emery manifolds (replacing equality by greater or equal to in the gradient Ricci soliton equation). Among a number of beautiful results, I should mention that they proved that shrinkers have at least linear volume growth (earlier stated by H.D. Cao).

This is just a selection of results; I'm sure I inadvertently omitted some I wanted to mention. By now, there are many more important papers in the field of Ricci solitons, perhaps too many to mention (I've omitted the Kaehler case; see for example Feldman--Ilmanen--Knopf for constructions with symmetry as Otis Chodosh mentioned as well as Dancer--Wang, Bo Yang, etal.; and the work of Xiaohua Zhu and coauthors in relation to Kaehler--Ricci flow). One just can search arXiv to see most of the recent papers.